\ 

LIBRARY 

' 


STAT. 


STAT. 


AN  INTRODUCTION  TO 
THE  LIE  THEORY  OF 
ONE-PARAMETER  GROUPS 

WITH    APPLICATIONS   TO 
THE   SOLUTION    OF   DIFFERENTIAL  EQUATIONS 


BY 

ABRAHAM    COHEN,   PH.D. 

ASSOCIATE   IN    MATHEMATICS,   JOHNS   HOPKINS    UNIVERSITY 


D.    C.    HEATH    &   CO.,    PUBLISHERS 

BOSTON         NEW   YORK         CHICAGO 
II  I 


BY  THE   SAME  AUTHOR 


AN    ELEMENTARY    TREATISE    ON 
DIFFERENTIAL   EQUATIONS 

ix  -f-  271  pages.     Half  Leather 


D.    C.    HEATH    &    CO..    PUBLISHERS 


COPYRIGHT,  1911, 
BY  ABRAHAM   COHEN. 

MATH. 

STAT. 

UBRARY 


STAT   • 

PREFACE 

THE  object  of  this  book  is  to  present  in  an  elementary  manner, 
in  English,  an  introduction  to  Lie's  theory  of  one-parameter  groups, 
with  special  reference  to  its  application  to  the  solution  of  differen 
tial  equations  invariant  under  such  groups. 

The  treatment  is  sufficiently  elementary  to  be  appreciated,  under 
proper  supervision,  by  undergraduates  in  their  senior  year  as  well 
as  by  graduates  during  their  first  year  of  study. 

While  a  knowledge  of  the  elementary  theory  of  differential  equa 
tions  is  not  absolutely  essential  for  understanding  the  subject 
matter  of  this  book,  frequent  references  being  made  to  places  where 
necessary  information  can  be  obtained,  it  would  seem  preferable  to 
approach  for  the  first  time  the  problem  of  classifying  and  solving 
differential  equations  by  direct,  even  if  miscellaneous,  methods  to 
doing  so  by  the  elegant  general  methods  of  Lie ;  and  this  book  is 
intended  primarily  for  those  who  have  some  acquaintance  with  the 
elementary  theory.  To  such  persons  it  should  prove  of  great  inter 
est  and  undoubted  practical  value.  An  attempt  has  been  made 
throughout  the  work  to  emphasize  the  role  played  by  the  Lie  theory 
in  unifying  the  elementary  theory  of  differential  equations,  by 
bringing  under  a  relatively  small  number  of  heads  the  various 
known  classes  of  differential  equations  invariant  under  continuous 
groups,  and  the  methods  for  their  solution.  Special  attention  may 
be  called  to  the  lists  of  invariant  differential  equations  and  applica 
tions  in  §§  19,  28,  30;  while  the  two  tables  in  the  appendix  include 
most  of  the  ordinary  differential  equations  likely  to  be  met. 

Only  as  many  examples  involving  the  solution  of  differential 
equations  as  seem  necessary  to  illustrate  the  text  have  been  intro- 


iv  PREFACE 

duced.     The  large  number  usually  given   in  the  elementary  text 
books  seems  ample  for  practice. 

The  short  chapter  on  contact  transformations,  while  not  essential 
to  the  work,  has  been  added  for  purposes  of  reference  and  to  give 
the  student  sufficiently  clear  ideas,  so  as  to  provide  a  working 
knowledge,  in  case  he  has  occasion  to  apply  them.  For  the  same 
reasons,  the  rather  sketchy  note  on  /--parameter  groups  has  been 
added,  where  an  attempt  is  made  to  bring  out,  as  concisely  as 
seems  consistent  with  clearness,  the  relations  between  r-parameter 
groups  and  their  infinitesimal  transformations.  An  exposition  of 
the  general  theory  would  be  beyond  the  scope  of  this  work. 

To  a  large  extent  Lie's  proofs  and  general  mode  of  presentation 
have  been  retained,  both  because  of  their  elementary,  direct  char 
acter,  and  because  the  subject  is  so  essentially  Lie's  own.  An 
attempt  has  been  made,  however,  at  a  more  systematic  arrange 
ment  of  the  subject  matter  and  at  identifying  more  closely  the 
classes  of  differential  equations  invariant  under  known  groups  with 
those  considered  in  the  elementary  theory. 

The  author  takes  pleasure  in  expressing  his  appreciation  of  the 
valuable  suggestions  made  by  Dr.  J.  R.  Conner,  who  kindly  con 
sented  to  read  the  proofs. 

ABRAHAM   COHEN. 

JOHNS  HOPKINS  UNIVERSITY, 

BALTIMORE,  MD.,  August,  1911. 


CONTENTS 

CHAPTER   I 

LIE'S  THEORY  OF  ONE-PARAMETER  GROUPS 

SECTION  PACK 

1.  Group  of  Transformations      .........  i 

2.  Infinitesimal  Transformation           * 6 

3.  Symbol  of  Infinitesimal  Transformation 8 

4.  Group  Generated  by  an  Infinitesimal  Transformation  10 

5.  Another  Method  of  Finding  the  Group  from  its  Infinitesimal  Transfor 

mation        ..       ^         .........  14 

6.  Invariants 16 

7.  Path-curves.     Invariant  Points  and  Curves 17 

8.  Invariant  Family  of  Curves 20 

9.  Change  of  Variables 23 

10.  Canonical  Form  and  Variables 26 

11.  Groups  Involving  More  than  Two  Variables 28 

CHAPTER   II 
DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER 

12.  Integrating  Factor 37 

13.  Differential  Equation  Invariant  under  Extended  Group           ...  40 

14.  Alternant 44 

15.  Another  Criterion  for  Invariance  of  a  Differential  Equation   under  a 

Group        ............  45 

16.  Two  Integrating  Factors 48 

17.  General  Expression  for  Group  under  which  a  Differential  Equation  is 

Invariant    .                                    49 

18.  Differential  Equations  Invariant  under  a  Given  Group  .         ...  50 

19.  Illustrations  and  Applications         ........  52 

20.  Second  General  Method  for  Solving  a  Differential  Equation.     Separa 

tion  of  Variables 63 

21.  Singular  Solution 66 

v 


vi  CONTENTS 


CHAPTER   III 

MISCELLANEOUS  THEOREMS  AND  GEOMETRICAL  APPLICATIONS 

SECTION  PAGE 

22.  New  Form  for  Integrating  Factor           .......  69 

23.  Two  Differential  Equations  with  Common  Integrating  Factor         .         .  72 

24.  Isothermal  Curves           ..........  72 

25.  Further  Application  of  the  Theorem  of  §  23          .....  76 


CHAPTER   IV 
DIFFERENTIAL  EQUATIONS  OF  THE  SECOND  AND  HIGHER  ORDERS 

26.  Twice-extended,  w-tiines-extended  Grbup       ......  83 

27.  Differential  Equation  of  Second  Order  Invariant  under  a  Given  Group  86 

28.  Illustrations  and  Applications         ........  90 

29.  Further  Applications      ..........  97 

30.  Differential  Equation  of  Order  Higher  than  the  ^econd  Invariant  under 

a  Given  Group  ...........       99 

CHAPTER   V 
LINEAR  PARTIAL  DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER 

31.  Complete  System  ...........  104 

32.  Method  of  Solution  of  Complete  System         ......  1  1  1 

33.  Second  Method  of  Solution   .........  113 

34.  Linear  Partial  Differential  Equation  Invariant  under  a  Group        .  115 

35.  Method  of  Solution  of  Linear  Partial  Differential  Equation  Invariant 

under  a  Group   ...........  119 

36.  Jacobi's  Identity     .........         .         .121 

37.  Linear  Partial  Differential  Equation  Invariant  under  Two  Groups          „  122 

38.  Methods  of  Solution  of  Linear  Partial  Differential  Equation  Invariant 

under  Two  Distinct  Groups        ........  124 

CHAPTER   VI 
ORDINARY  DIFFERENTIAL  KQI-ATIONS  OF  THK  SITOND  ORDER 

39.  Differential  Equation  of  the  Second  Order  Invariant  under  a  Group     .     134 

40.  Differential  Equation  of  the  Second  Order  Invariant  under  Two  Groups     137 

41.  Other  Methods  of  Solution     .........      142 


CONTENTS  vii 

SECTION  PAGE 

42.  Number  of  Linearly  Independent  Infinitesimal  Transformations   that 

Leave    a    Differential    Equation    of    the    Second    Order    Unaltered, 

Limited 143 

43.  r-parameter  Group  of  Infinitesimal  Transformations       ....  146 

44.  Classification  of  Two-parameter  Groups 152 

45.  Canonical  Forms  of  Two-parameter  Groups 155 

46.  DiftVrential  Equation  of  the  Second  Order  Invariant  under  Two  Groups  165 

47.  Second  Method  of  Solution  for  B 169 

CHAPTER  VII 
CONTACT  TRANSFORMATIONS 

48.  Union  of  Elements 175 

49.  Contact  Transformation 178 

50.  Group  of  Contact  Transformations.     Infinitesimal  Contact  Transforma 

tion    .         .         •       *• •         •         •         •  l85 

51.  Ordinary  Differential  Equations 189 

52.  First  or  Intermediary  Integrals 191 

53.  Differential  Equation  of  the  First  Order  Invariant  under  a  Group  of 

Contact  Transformations    .         .         .         .         .         .         .         .         .194 

APPENDIX 

Note  I. — The  Infinitesimal  Transformation 197 

Note  II.  —  Solution  of  the  Riccati  Equation  of  §  18 201 

Note  III.  —  Isothermal  Curves 203 

Note  IV.  —  Differential  Equation  of  the  Second  Order  not  Invariant  under 

Any  Group 206 

NoteV.  —  (t71'C7«')/=(t7iUzy/ '  .  .209 

Note  VI.  —  Continuous  Groups  Involving  More  than  One  Parameter  .  .211 

Note  VII.  —  Condition  for  Essential  Parameters 226 

Table  I  —  Differential  Equations  of  the  First  Order  Invariant  under  Known 

Groups 231 

Table  II.  —  Differential  Equations  of  Higher  Order  Invariant  under  Known 

Groups 236 

ANSXVKKS  TO  EXAMPLES 239 

INDEX „ 247 


LIE'S   THEORY   OF 
DIFFERENTIAL    EQUATIONS 

CHAPTER   I 
LIE'S  THEORY  OF  ONE-PARAMETER  GROUPS 

1.   Group  of  Transformations.  —  The  set  of  transformations 
(i)  xl  =  <l>(xlyt  a),  yi  =  \l/(x,y,  a)* 

each  one  being  determined  by  some  value  of  the  parameter  a,  con 
stitutes  a  group  if  the  transformation  resulting  from  the  successive  per 
formance  of  any  two  of  them  is  one  of  the  transformations  of  the 
aggregate.  In  other  words,  assigning  a  definite  but  arbitrarily  selected 
value  to  the  parameter  a,  and  then  any  second  value  b  (where  b  may 
or  may  not  be  equal  to  a),  this  second  transformation  being 

(i6)  ,r2  =  4>(xlt  ylt  &),  y,  =  *l/(xlf  y},  6), 

the  transformations  of  type  (i)  form  a  group  if  the  results  of  eliminat 
ing  Xi  and  j't  from  (i)  and  (i6),  i.e. 

,yt  a),  \l/(x,y,  a\  ^]»  y-2 


*  Here  0  and  \f/  are  supposed  to  be  generally  analytic,  real  functions  of  the  three 
quantities  xty,  a;  and,  unless  especially  stated,  it  will  be  understood  that  .r  and  y  are 
real,  and  that  a  takes  such  values  only  as  render  .rj  and  y^  real.  Besides,  0  and  \J/  are 
independent  functions  with  respect  to  .r  anfi  y,  alone;  i.e. 

(90     (>/> 


_r    dy 

so  that  equations  (i)  can  be  solved  for  .1  and  y. 


2  THEORY   OF   DIFFERENTIAL    EQUATIONS  §i 

reduce  identically  to 

x2  =  <j>(x,  y,  c),  y2  =  $(x,  y,  c\ 

where  c  is  a  function  of  a  and  b  only.  If  (i)  be  represented  by  Tn 
and  (i6)  by  Tb,  the  group  property  may  be  expressed  symbolically 

TaTb=Tc. 

We  shall  speak  of  TaTb  as  the  product  of  Ta  and  Tb ;  and  shall  under 
stand  that  it  represents  the  transformation  resulting  from  the  succes 
sive  performance  of  Ta  and  Tb,  in  the  order  named.  With  this  in 
mind,  the  group  property  of  a  set  of  transformations  may  be  expressed 
in  the  words,  the  product  of  any  two  transformations  of  the  group  is 
equal  to  some  transformation  of  the  aggregate. 

As  an  example,  consider  the  translations* 

I  x\  —  x,  yi  —  y-\r  a. 

After  having  fixed  upon  some  value  a  of  the  parameter,  a  second  transformation 
of  the  set,  corresponding  to  the  value  l>,  is 

•*a  =  *\i  )'-2  =-y\  +  b- 

The  result  of  the  successive  performance  of  the  two  is 
x2  =  x,  y*  =  y  +  a  +  l>, 

which  is  again  a  translation  of  the  set,  with  a  -f  />  as  the  value  of  the  parameter. 
Hence,  all  translations  of  the  type  I  form  a  group. 
As  another  example,  consider  the  rotations^ 

II  Xi  =  x  cos  a  —  y  sin  a,    y\  =  x  sin  a  +  y  cos  a. 

*  It  will  frequently  be  found  convenient  to  consider  this  subject  from  a  geometrical 
point  of  view.  A  transformation  of  the  form  (i)  may  be  looked  upon  as  transforming 
the  point  (x,y)  into  the  point  (TJ,;',).  The  effect  of  a  transformation  I  is,  obviously, 
to  carry  any  point  the  distance  ,i  in  the  direction  of  the  axis  of  r.  So  that  the  effect  on 
all  the  points  of  the  plane  is  that  of  a  translation  of  the  whole  plane  over  a  distance  a 
in  the  dire'etion  of  the  axis  of  v. 

f  Obviously  the  effect  of  a  transformation  of  this  type  on  the  various  points  of  the 
plane  is  that  of  a  rotation  of  the  whole  plane,  through  the  angle,  a,  about  the  origin. 


§i  THKORY   OK   <>NTK- PARAMETER   GROUPS  3 

The  result  of  first  performing  the  transformation  corresponding  to  some  definite 
value  of  a,  and  then  a  second  one, 

x-z  =  xi  cos  b  —  yi  sin  b,  y*  =  xi  sin  b  -j-  y\  cos  b, 

is    x2  =  x  cos  (a  +  /;)  —7  sin  (a  +  £),      y»  =  x  sin  (a  +  /;)  +y  cos  (<i  +  <$), 
which  is  again  a  rotation  of  the  set,  with  a  +  ^  as  the  value  of  the  parameter. 

Hence,  all  rotations  of  the  type  II  form  a  group. 

The  ajfine*  transformations, 

III  xi  =x,  yi=  ay, 

form  a  group,  since  the  result  of  two  transformations  in  which  the  values  of  the 
parameter  are  a  and  l>,  respectively,  is 

•*2  =  •*»  y*  —  a  h', 

where  ab  is  the  value  of  the  parameter. 

In  the  same  way  it  is  readily  seen  that  the  perspective  or  similitudinous  f 
transformations, 

IV  .  xi  =  ax,  yi  =  ay, 
form  a  group. 

In  the  groups  considered  in  the  Lie  theory  it  is  presupposed  that 
the  transformations  can  be  arranged  in  pairs,  the  members  of  which 
are  mutually  inverse  \  ;  that  is,  if  (i)  be  solved  for  x  and  y,  their 
values  in  terms  of  x^  and  y\  assume  the  forms 

(i)  x  =  <K*i,  y\,  a),  y  =  «AOi>  y\>  «)> 

where  a  is  some  function  of  a. 

Thus  in  the  examples  above  we  have  the  inverse  transformations  : 
I.  x  =  xi,  y=yl  —  a;  here          a  =—  a. 

II.          x  =  xi  cos  a  •}- y\  sin  a,  y  =  —  x\  sin  a  +  y\  cos  a  ;   a  = — a. 

III.  x  =  x\,  y  =  ~y-l-i 

a 

iv.       *  =  !jrlf  y  =  *y\\  *  =  - 

a  a  a 

*  Following  Lie,  this  name  is  used  here  in  a  restricted  sms<>  to  npply  to  transforma 
tions  of  the  types  III  and  III',  $  19.  The  term  goes  back  to  Moebius  (1790-1868), and 
usually  includes  all  entire  linear  transformations  Xi  =<Zj  x  +  b\y  -\-Ci,yi~<i-2%~\~  b^y  +  ^3. 

f  So  called  because  the  effect  of  any  one  of  them  is  to  stretch  the  vector  going  from 

the  origin  to  the  point  (.r,_y)  in  the  ratio  (I ,  leaving  its  direction  unaltered.     Any  figure 

I 
in  the  plane  is,  therefore,  transformed  into  one  similar  to  it  by  a  transformation  IV. 

J  Such  groups  will  be  referred  to  as  Lie  groups  when  this  property  is  to  be  em 
phasized. 


4  THEORY   OF   DIF1-  KRKN  TIAL    EQUATIONS  §  i 

Since  the  successive  performance  of  two  mutually  inverse  trans 
formations  results  in  the  identical*  transformation,  the  latter  must 
always  be  a  transformation  in  every  group  considered  in  this  theory  f  ; 
hence,  there  must  always  exist  a  value,  a(),  of  the  parameter  which 
reduces  the  corresponding  transformation  to  an  identity 

,)  a0)=x, 


It  is  readily  seen  that  in  the  case  of  I,  II,  III,  IV  the  values  of  «0  are  o,  o,  I,  I 
respectively. 

Since  <£  and  if/  are  continuous  functions  of  the  parameter  a,  if  we 
start  with  the  value  aQ,  and  allow  a  to  vary  continuously,  the  effect 
of  the  corresponding  transformations  on  x  and  y  will  be  to  transform 
them  continuously  ;  that  is,  for  a  sufficiently  small  change  in  a  the 
changes  in  x  and  y  are  as  small  as  one  pleases.  Looked  at  geometri 
cally,  the  effect  will  be  to  transform  the  point  (x,  y)  to  the  various 
points  on  some  curve,  which  is  known  as  a  path-curve  of  the  group. 

Thus  in  the  case  of  I,  the  point  (x,  y)  is  transformed  into  the  various  points 
on  the  line  through  it,  parallel  to  the  axis  of  y\  in  the  case  of  II,  the  path-curves 
are  obviously  circles  having  the  origin  for  center  ;  in  III  the  path-curves  are 
again  lines  parallel  to  the  axis  of  y,  while  in  IV  the  path-curves  are  straight  lines 
through  the  origin. 

It  is  evident  that  when  x  and  y  are  considered  as  constants  while 
xl  and  yl  are  taken  as  variables,  the  equations  (i)  are  the  parametric 
equations  of  the  path-curve  through  the  fixed  point  (x,  y).  Hence, 
the  path-cuwe  corresponding  to  any  point  (x,  y)  may  be  obtained  by 
eliminating  a  from  the  two  equations  of  (\). 

*  Identical  transformation  is  the  name  given  to  a  transformation  that  leaves  un 
altered  all  the  elements  upon  which  it  operates. 

\  Groups  exist  in  which  the  parameter  enters  in  such  a  way  that  there  is  no  iden 
tical  transformation.  (S.-«-  Lie,  I'ran.<;f<>ni/ati(»i\!;riif>pen,Vo\.\<{4<\.)  Such  groups 
will  not  be  included  among  those  considered  here. 


§i         THEORY  OF  ONE-PARAMETER  GROUPS          5 

Remark  i.  —  It  is  readily  seen  that,  in  general,  the  path-curve 
corresponding  to  any  point  corresponds  equally  well  to  every  other 
point  on  it. 

There  is  a  possible  exception  to  this  statement.  A  point  may  be  left  un 
altered  by  every  transformation  of  the  group;  as,  for  example,  the  origin  in  the 
case  of  II.  Such  a  point  would  naturally  not  have  a  path-curve.  In  the  case  of 
III,  every  point  on  the  axis  of  x  is  left  unaltered;  hence,  a  line  parallel  to  the 
axis  of  y  is  the  path-curve  of  every  point  on  it,  except  the  point  where  it  cuts  the 
axis  of  x.  In  IV  a  line  through  the  origin  is  the  path-curve  of  every  point  on  it, 
except  the  origin,  which  is  left  unaltered. 

Remark  2.  —  The  parameter  may  appear  in  various  forms  in  the 
transformations  that  determine  a  given  group. 

Thus  x\  —  x,  y\  —y  +  a1  also  determines  the  group  of  translations  I.  In  this 
case  a  must  take  imaginary  values,  as  well  as  real  ones,  in  order  to  give  all  the 
transformations  of  I.  As  a  matter  of  fact  a  =  ia.  On  the  other  hand,  a  negative 
value  for  a  determines  the  same  transformation  as  the  corresponding  positive 
value. 

The  group  of  rotations  II  can  also  obviously  be  written 

x\  —  x  Vi  —  a1  —  ya,  y\  —  xa  +  yVi  —  a1. 

It  is  always  possible  (and  in  an  indefinite  number  of  ways)  to 
choose  as  a  new  parameter  such  a  function  of  the  parameter  appear 
ing  in  any  group  that  the  value  giving  the  identical  transforma 
tion  is  any  desired  number.  For  example,  this  number  will  be  0  if 
a  is  replaced  by  a^e'1'0.  In  particular  it  will  be  zero  if  a  is  replaced 

by  a«e". 

« 

Thus  if  III  and  IV,  where  at)  =  I,  are  written 

•*"!  =  x,  y\  —  eay  and  x\  —  e°x,  y\  =  e*y, 

respectively,  a  =  o  will  determine  the  identical  transformation.  In  this  form, 
complex  values  of  a  are  necessary  to  determine  transformations  which  cor 
respond  to  negative  values  of  the  parameter  in  the  original  forms  of  the  trans 
formations  of  these  groups. 


6  THEORY    OF   DIFFERENTIAL    EQUATION'S  §§  I,  2 

Show  that  the  following  transformations  constitute  a  group.  Find 
the  respective  values  of  the  parameter  that  give  both  the  inverse  and 
the  identical  transformations.  Also  find  the  path-curves  :  — 

Ex.    1.    xi  =  ax,  Vi  =  -y.      Ex.    2.    x{  —  a~x,  }\  =  ay. 
Ex.    3.   A\  =  a'2x,  )\  =  a*y. 


Ex.  4.    xl  =  +     .v"  +  2  a,  }\  =  -\-  vV  —  a. 

Ex.  5.    A*!  =  x  cosh  a  +y  sinh  a,yi  =  xs\nha  -\~y  cosh  a. 

Ex.   6.    ^,= 


i  —  ax   '         i  —  ax 
Ex.   7.    .T!  =  ax-\-(a—  i)j',  ,i'i=J. 
Ex.  8.   #!  =  ^a(.r  cos  d!  —  y  sin  «),' ji  =  e\x  S\T\a-\-y  cos  a). 

2.    Infinitesimal  Transformation. — Since  ^>  and  ^  are  continuous 
functions,  the  transformation 

x\  =  4>(x,  y,  0o  +  8«),  y\  =  ^(x,  y,  «0  +  &?), 

where  AO  is  the  value  of  the  parameter  determining  the  identical  trans 
formation  and  &a  is  an  infinitesimal,  changes  x  and  y  by  infinitesimal 
amounts.  Developing  by  Taylor's  Theorem 


Noting  that  <#>(.r,>',  «0)=^  ^(^^»  ^o)  =  J,  the  chang'es  in  x  and 
due  to  the  transformation  are 


§2  THEORY   OF  ONE-PARAMETER   GROUPS  / 

where  terms  in  higher  powers  of  $a  are  indicated  by  dots.     Since  aQ 
is  a  fixed  value  of  the  parameter,  the  only  variables  remaining  in 

and  f^\     are  x  and  y.     Writing 

>• 

the  transformation  takes  the  form 


Higher  powers  of  the  infinitesimal  8a  may  be  neglected,  provided  at 
least  one  of  £  and  77  does  not  vanish  identically  (i.e.  for  all  values  of 
x  and  y),  and  neither  of  them  is  infinite.  In  this  case  the  transforma 
tion  producing  an  infinitesimal  change  in  the  variables  is 

(2)  8*  =  £(.*,jO&i,    Sy  =  r,(x,y)?>a. 

This  is  known  as  an  infinitesimal  transformation. 

Remark  i.  —  Sinoe  k$a,  where  k  is  any  finite  constant  different 
from  zero,  is  an  infinitesimal  when  £z  is,  the  latter  may  be  replaced 
by  the  former  in  (2).  Hence,  the  infinitesimal  transformation  (2)  is 
the  same  as  n,  ,«./ 


On  -the  other  hand,  if/(.r,  v)  is  not  a  constant, 

&r  =  /(*,  y)  •  ^,  y)Sa,    8v  =/(.v,  v)  -  ^ 
is  distinct  from  (2). 

Remark  2.  —  In  case  [    *]     and  f    ^]     are  both  identically  zero, 


or  if  one  of  them  is  infinite,  the  method  of  this  section  for  finding  an 
infinitesimal  transformation  of  the  group  must  be  modified.  In  Note  I 
of  the  Appendix  the  existence  of  an  infinitesimal  transformation  of 
the  group  is  established  in  every  case,  and  a  method  for  finding  it  is 
also  given.  Moreover,  in  the  same  note  it  is  proved  t\\-\\  a  one  -param 
eter  group  contains  only  one  Distinct  infinitesimal  transformation. 


8  THEORY  OF   DIFFERENTIAL   EQUATIONS  §§  2,  3 

In  general,  the  method  of  this  section  for  finding  £  and  77  will  be 
found  applicable  ;  when  not,  that  of  Note  I  may  be  employed. 

In  the  case  of  I  the  infinitesimal  transformation  is 

•*i  =  x,  y\  =  y  +  8a, 
or  dx  =  o,    dy  =  8a  ; 

f=o,  ,=,.      r3*=ftf*=,.i 

Lda  da          J 

For  II,  the  infinitesimal  transformation  is 

x\  =  x  cos  (da)  —  y  sin  (da),  y\.—x  sin  (da}  +  y  cos  (da}. 


Since   cos(5«)  =  i  -  +  ...,  and  sin  (5«)  =  $a  —  +  ...,    and    infini 

tesimals  of  higher  order  than  the  first  may  be  neglected,  cos(Srf)  may  be  re 
placed  by  i,  and  sin  (8a)  by  da.  Hence, 

dx  =  —  yda,    dy  =  xda  ; 

«—  *    '-* 

Similarly,  it  is  readily  seen  that  for 

III  £  =  o,    i=y, 

IV  l  =  x,    r)=y. 

Ex.     Find  the  infinitesimal  transformations  of  the  groups  in  the 
exercises  of  §  i. 

3.    Symbol  of  Infinitesimal  Transformation.  —  In  the  infinitesimal 
transformation 
(2)  &x  =  t(x,y)Ba,  8y  =  y(x,  y)Ba, 

8  is  the  symbol  for  differentiation  with  respect  to  the  parameter  a  ; 
but  in  a  restricted  sense,  since  it  is  used  to  designate  the  value 
which  the  differential  of  the  new  variable  x^  or  \\  assumes  when  a  = 
a0*  Thus 


*  The  exceptional  cases  noted  in  Remark  2,  $  2  are  due  to  the  way  in  which  the 
parameter  enters  and  are  not  peculiar  to  any  .qrmip.  (See  §  4.)  Hence.no  modifica 
tion  of  the  statement  made  in  the  text  need  be  insisted  upon,  provided  it  is  understood 
that  the  parameter  is  chosen  in  proper  form. 


§3  THEORY   OF  ONE-PARAMETER   GROUPS  9 

If/(jc,  y)  is  a  generally  analytic  function  of  x  and  y,  the  effect 
of  the  infinitesimal  transformation  on  it  is  to  replace  it  by 
/(#  +  £&*,  y+rj&a),  which  on  expanding  by  Taylor's  Theorem 
becomes 


x         dy 
Hence,  y 


Lie  introduced  the  very  convenient  symbol  6^  for  the  coefficient 
of  Ba  in  this  expansion  ;  so  that 


where 
(3)  #S« 

It  is  readily  seen  that         6^=  / 


where  A  =-/(xlf  yj  . 

In  particular  C^c  =  ^,    CJ'  =  >;. 

Since  Uf  can  be  written  when  the  infinitesimal  transformation  (2) 
is  known,  and  conversely,  (2)  is  known  when  Uf  is  given,  Uf  is  said 
to  represent  (2).  For  convenience  of  language  we  shall  usually 
speak  of  "  the  infinitesimal  transformation  Uf"  instead  of  "  the  trans 
formation  represented  by  Uf"  But  it  must  be  borne  in  mind  that 
Uf  is  nota.  transformation;  it  is  only  the  representative  of  one. 

The  infinitesimal  transformations  in  the  cases  of  I,  II,  III,  IV  are 


respectively. 

Remark.  —  The  differential  operator  £7=  £ 1-  -n  —  has  striking 

av          or 

properties,  many  of  which  will  be  brought  out  in  the  course  of  this 


10  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§3,4 

work.  It  is,  to  a  large  extent,  because  of  these  properties  that  Lie's 
introduction  of  the  idea  of  the  infinitesimal  transformation  has 
proven  so  prolific  of  results. 

4.  Group  Generated  by  an  Infinitesimal  Transformation.  —  In  §  2 
was  given  a  method  for  finding  the  infinitesimal  transformation  of  a 
one-parameter  group  when  the  finite  transformations  are  given.  Con 
versely,  the  finite  transformations  can  be  obtained  when  the  infini 
tesimal  transformation  is  known. 

Attention  was  called  in  Remark  2,  §  i,  to  the  fact  that  the 
parameter  may  be  made  to  enter  in  such  a  way  that  the  identical 
transformation  is  given  by  any  desired  value  of  the  parameter.  It 
is  frequently  convenient  to  have  the  parameter  in  such  a  form 
that  its  vanishing  gives  the  identical  transformation.  In  future, 
when  this  is  specifically  understood,  /  will  be  used  for  the  param 
eter.  In  the  general  case,  when  this  form  is  not  insisted  upon,  a 
will  be  retained. 

The  infinitesimal  transformation  £7=£  —  +  n  — ,  or 

dx         dy 

(2')  &r  =  ^,^)8/,    8j'  =  ^,j-)8/, 

carries  the  point  (.r,  r)  to  the  neighboring  position  (x  -\-  &t, 
y  +  rj8/).  The  repetition  of  this  transformation  an  indefinite 
number  of  times  has  the  effect  of  carrying  the  point  along  a  path  * 
which  is  precisely  that  integral  curve  of  the  system  of  differential 
equations 

(4)  5  =  <(*,,J',),   f  =  *<•*«•.>•')• 

which  passes  through  the  point  (x,  /).  At  any  stage  of  the  above 
process  x  and  y  have  been  transformed  into  xl  and  jr,,  and  the 

*  This  is  obviously  tin-  path-curve  (^  i)  of  the  group,  corresponding  to  the  point 
(*,>). 


§4  T1IKORY   OK   ONE-PARAMETER   GROUPS  II 

formulae  of  transformation  are  given  by  those  solutions  of  (4)  or  of 
their  equivalents 


for  which  Xi  reduces  to  x  and  y\  to  j  for  /=  o. 

The  first  two  members  of  (5)  being  free  of  /  form  a  differential 
equation  whose  solution  may  be  written 


since  x\  =  x,  y\=y  when  /=o.      This  is  the  equation  of  the  path- 
curve  corresponding  to  the  point  (xt  y). 

Solving  ufa,  }>i)  =  c  for  one  of  the  variables,*  to  fix  the  idea,  say 
jfl  =  o>(v1,  c\  and  replacing  xl  in  77  by  to,  the  resulting  differential 

equation  », 

^- =  dt 

can  be  solved  by  a  quadrature.     Replacing  c  by  its  value  in  terms  of 
xl  and  )\  this  solution  takes  the  form 

v(xlt  }\)  -  t  =  const.  =  v  (x,  y). 
Hence  it  follows  that 
(6) 


determine  x^  and  y\  as  those  solutions  of  (4)  or  (5)  which  reduce  to 

x  and  y  respectively  for  /  =  o. 

Looking  upon  (6)  as  a  transformation,  the  following  may  be  noted  : 
i°   The  result  of  the  successive  performance  of  two  transformations 

U(xlt  _v,)  = //(.v,  v)        1  |  //(.v,,  v,")    :  ;/(.v,.  r, ), 

\  v(X'>,  y->)=  yu-,.  .r,  >  4-  /', 

*  At  limes  it  \\ill  l>e  more  practical  to  uso  somt>  of  tlie  other  methods  given  in 
the  author's  Klrmrnt;,!  v  Treatise  on  I  Mftcivntial  I '.(piations  (in  future  referred  to  as 
Kl.  1  >it.  Eq.)  §  65  for  finding  a  second  solution  of  (5). 


12  THEORY  OF   DIFFERENTIAL   EQUATIONS  §4 

is  the  same  as  that  of  the  single  transformation 


2°  The  value  —  /  determines  the  transformation  inverse  to  that 
obtained  by  using  /. 

3°    t=o  gives  the  identical  transformation. 

Hence  the  aggregate  of  all  the  transformations  (6)  for  all  values 
of  /  constitute  a  group  of  the  kind  considered  in  the  Lie  theory  (  §i). 
This  group  (either  in  the  form  (6)  or  when  solved  for  xl  and  j,)  is 
known  as  the  group  generated  by  the  infinitesimal  transformation  (2').* 

Moreover,  the  parameter  enters  in  such  a  way  that  f      l )  =  £(x,  y), 
~  \  =  r^x,  y).     Since  there  was  no  restriction  placed  on  the  £  and 

77  in  (2'),  other  than  that  they  are  generally  analytic,  which  is  always 
presupposed,  we  have  shown  that  it  is  always  possible  to  put  the  finite 
transformations  of  a  one-parameter  group  in  such  form  that  the  ex 
ceptional  cases  noted  in  Remark  2,  §  2  will  not  arise. 

dx\      dv\      dt 

In  1.  equations  ( O  are  —  =  -^— -  =. — • 

o  i         I 


v(*\>y\)=y\  -y  +  '• 

In  II,  dxL  =  </y1  =  df 

-  y\     *\      i 
. •.  «Oi, yi) = xi2  +  }>i*  =  x'~  -f  y2. 

Using  method  3° (a)  of  §  65,  El.  Dif.  Eg. 


•  '•  v(x,y)=tan'}-  '  =  tan"1"    -\- t. 

X  |  .V 

*  Since  the  finite  transformations  of  a  group  can  be  calculated  when  its  infinitesi 
mal  transformation  is  known,  the  latter  may  be  looked  upon  ;is  the  representative  of 
the  Ljroup.  We  shall  often  speak  of  "  the  group  ITf"  understanding  by  this  "  the  group 
whose  infinitesimal  transformation  i^  lepresented  by  Uf." 


§4  TIIKORY  OF  ONE-PARAMETER   GROUPS  13 

To  solve  these  two  equations  for  *i  and  y\,  so  as  to  obtain  the  transformation 
in  the  usual  form,  one  may  proceed  as  follows  : 

Taking  the  tangent  of  each  side  of  the  second  relation, 

i'i      y  4-  .YT 

—  =  —     — ,  where  r  =  tan  /. 

xi      x  —  yr 

Adding  I  to  the  square  of  each  side  and  taking  account  of  the  first  relation, 


Whence  x\  =  —         —  =  xcost  —  y  sin  /, 

Vl+T* 

—  =  cos  /  and  =_  =  sin  /; 


Vi  +  T2  Vi  +  T'< 

XT  +   V 

and  Vi  =  —         =  =  x  sin  /  +  y  cos  /. 

Vi    +   T'2 

In  III,  it  is  readily  seen  that 

*(JT1,  /l)  =Xi  =  X, 

v(*i,  vi)  =\ogyi  =  log^  +  /,  oryi  =  *y. 

Note.  — It  is  evident  that  the  solutions  of  (5)  need  not  always  be  found  in  the 
form  (6).     Other  forms  may  be  easier  to  solve  for  x\  and  y\.     Thus  in  IV 

log  x\  —  log  x  +  I  and  log^i  =  log  y  +  / 

are  a  pair  of  obvious  solutions  of  the  differential  equations  (5),  and  lead  at  once  to 

*\  =  *xt  y\  =  <*y. 

Find  the  groups  whose  infinitesimal  transformatiohs  are  the  follow 
ing  : 

Ex.   1.   4^-"r-  Rx-   5-    *¥  +  *¥• 

ox     "  oy  ox         oy 

Ex.2.    2.1-f+jf.  Ex.6,    .v-'f  +  .vrf. 

3.v         dy  dx  .     -   dy 

Ex.3.    2.rg+3J.|.  Ex.7.    (.v+.v)f- 

Ex.   4.     'f-^f-  Ex.8.    (,-,.)f+(.v+>4. 

.\  (U-      2V  dy  *'&»  -    dy 


14  THEORY    OF    DIFI'ERENTIAL    EQUATIONS  §5 

5.    Another  Method  of  Finding  the  Group  from   its  Infinitesimal 
Transformation.  —  Startin    with  an  infinitesimal  transformation 


dy 
it  was  seen  in  §  4  that  the  finite  transformations  of  the  group 

(i')  *i  =  <l>(x,y,  t),y\  =  t(x,y,  /) 

generated  by  it  can  be  found  in  such  form  that 


The  finite  transformations  can  be  obtained  (expanded  in  powers 
of  /)  without  integration  by  means  of  the  following  considerations  : 

The  effect  of  any  transformation  (V)  being  to  replace  x  and  y  by 
xl  and  yly  it  will  change  any  function  /(x,  y)  into  /(xlt  }\).  Assum 
ing/^,  j)  to  be  generally  analytic,  since  f(xlf  ji)  depends  upon  /  it 
can  be  developed  by  Maclaurin's  Theorem. 


where  f=f(x,  y),A  =/(»i,  2/i).     Writing  likewise 


so  that  (^)0  =  £  (^o  =  77,  (  C/j/Oo  =  Uf,  it  follows  that 

=UJ,  whence  W 


Moreover  =      ^7,  =  ^  (7^=  U&. 

Hence      f^A  =  UUf= 


Similarly  =  UUUf=k  U*f;  and  so  on.     Hence  the  effect  of 


§5  THEORY   OF  ONE-PARAMETER  GROUPS 

any  finite  transformation  (V)  on /is  given  by 

(7)  /i 


In  particular  the  finite  transformations  of  the  group  are  given  by 
the  formula  (7)  when /is  simply  x  and  y,  thus 

t* 
(8)  *^\ 

t 

2  I 

where,  it  will  be  recalled  (§  3),  Ux  =  £,  Uy  =  rj. 

It  is  readily  seen  that  for  the  group  in  the  form  (8)  as  in  the  form 

In  I          Uf=&. 

Ux  =  o,  U^x  -  o,  •••  ;    Uy  =  i,  U~y  —  o,  U'Ay  =  o, .... 
Hence  x\  =  x,  y\  =  y  -f-  /. 

In  II  Uf=-y&+x& 

Qjc         {jy 


Ux  =  -y,  U*x  =  U(-y}  =-  x,  U*x=  U  (-  x)=  yt  and  so  on. 
_  .<!  +  '1  _...)_>(,_/!  +  i  _...) 

2!    4!       /      V     3!    51       / 


Hence 


—  X  COS / — 

Similarly    y±  = 

=  x  sin  /  -f  y  cos  /. 
In  III         Uf=y  §/-m 

dy 

Ux  —  o,  U*x  -  o,  ...  ;    Uy  -y,  U*y  =  y,  U*y 

Hence  x\.  —  x, y\_  =  y(  i  +  /  +  /-  -|-  r     f  ...) 

'V  2!      3!  / 

*  Symbolically  this  may  be  written    /",  _  tf('f. 


l6  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§5,6 

In  exactly  the  same  v  ay  the  finite  transformations  of  IV  are  found  to  be 
xi  =  xel,yi  =  }>(•'. 

Ex.    Solve  the  problems  of  §  4  by  the  method  of  this  section. 

6.  Invariants.  —  A  function  of  the  variables  is  said  to  be  an  inva 
riant  of  a.  group  (or  invariant  under  the  group)  if  it  is  left  unaltered 
by  every  transformation  of  the  group. 

Thus,  it  is  immediately  obvious  that  any  function  of  x  alone  is  invariant  under 
the  groups  I  and  III,  while  any  function  of  x'1  +/'  is  such  under  II. 

We  saw  (§5)  that 

(7)  A*i,  *)-/(*,  ;')  -  W+  wy^  +  .... 

In  order  that  f(xlf  )i)=/(-v,  y)  for  all  values  of  x  and  v,  and  the 
corresponding  values  of  xl  and  y\  into  which  they  are  transformed  by 
each  of  the  transformations  of  the  group,  i.e.  for  every  value  of  /,  it 
is  necessary  and  sufficient  that  each  coefficient  in  the  right-hand 
member  of  (7)  be  zero  for  all  values  of  x  and  y.  in  particular,  it  is 
necessary  that 


Moreover,  since  U~f=UUf,  U"f=UU~f,  •••,  it  follows  at  once 
that  (9)  is  also  the  sufficient  condition  that  f(x\,  y\)=f(xt  y)  for  ail 
values  of  x,  y,  and  /.  Hence,  the 

THKORKM.  —  The  necessary  and  sufficient  condition  1hatf(xty)  be 

invariant  under  flic  group  I  /  is  L'f=  o. 

« 
Remark.  —  This  theorem  may  also  be  expressed  as  follows:    77/r 

necessary  and  sufficient  condition  that  f-.\\  \}  he  invariant  under  a 
one-parameter  group  is  tJiat  it  he  left  unaltered  h\  the  infinitesimal 
transformation  of  the  group.  On  succeeding  pages  will  be  found 
conditions  for  invariance  of  curves,  families  of  curves,  differential 
equations  of  various  types,  and  so  on.  In  each  case  it  will  be  found 


§§6,7  THEORY   OF  ONE-PARAMETER   (IKOUl'S  17 

(although  specific  mention  of  the  fact  will  not  be  made)  that  the  con 
dition  for  invariance  under  the  group  always  reduces  to  that  of  in- 
variance  under  the  infinitesimal  transformation  of  the  group. 

To  determine  invariant  functions,  it  is  necessary  to  solve  the  partial 
differential  equation  •)  /•        ^f 


The  corresponding  system  of  ordinary  differential  equations  is 
(10)  ^  =  ^  =  tf. 

t  1J  O 

f=  const,  is  one  solution  of  the  system. 

If,  besides,  u(x,  y)=  const,  is  the  solution  of  the  equation  involving 
the  first  two  members,  the  general  solution-  of  (9)  is,  by  Lagrange's 
method,* 


In  I  and  III 

o        77        o 

.'.   H  =  JC\  and/=  l<\x). 
In  II  u~x^  -f/2  ;  and/=  F(. 

In  IV  «  =  ^;  and/= 


Ex.    Find  the  invariants  of  the  groups  in  the  problems  of  §  4., 

7.  Path-curves.  Invariant  Points  and  Curves.  —  As  was  seen  in 
$  4,  the  differential  equation  of  the  path-curves  of  a  group  is  readily 
obtained  from  the  infinitesimal  transformation  of  the  group.  Thus, 
u.iing  .v  and  v  as  the  variables,  it  is 

dy  _  rj 


(ii)  =      . 

t        t] 

*  See  El.  Dif.  Eq.  \  79. 


1  8  THEORY  OF   DIFFERENTIAL    EQUATIONS  §7 

The  general  solution  of  this  equation, 

u  (x,  y)  =  const., 

is  the  equation  of  the  family  of  path-curves.  As  u  (x,y)  is  an  invariant 
of  the  group  (§  6),  it  follows  that  the  equation  of  a  path-curve  is 
obtained  b\  equating  an  invariant  to  a  constant.  Moreover,  it  is  clear 
that  this  property  is  characteristic  of  an  invariant  ;  that  is,  if  equa 
ting  a  function  to  any  constant  whatever  gives  the  equation  of  a 
path-curve,  that  function  must  be  an  invariant. 

But  this  is  not  the  only  form  in  which  the  equation  of  a  path- 
curve  may  appear.*  A  path-curve  is  an  invariant  curve  of  the 
group,  hence  its  equation  must  be  invariant.  \if(xt  y)  =  o  is  to  be 
an  invariant  equation,  f(x^  ji'j)  must  vanish  for  all  values  of  x±  and  ft 
into  which  the  various  values  of  a:  and  y  which  satisfy  y"(  jc,  jj1)  =  o  are 
transformed  by  the  transformations  of  the  group.  Now,  we  have  seen 


(?)  /(-*•„  JO  =/(*,  y)  + 

If  the  right-hand  member  is  to  vanish  whenever/^,  y)  does,  for  every 
value  of  /,  it  is  necessary  and  sufficient  that  each  coefficient  should  do 
so.  *In  particular,  it  is  necessary  that 

(12)  Uf=  o,  whenever  f(xt  y)  =  o, 

that  is,  £^must  contain  f(xty)  as  a  factor.  | 

But  if  Uf=<*(x,y)f(x,y), 

then  Wf  =  UUf=  Uuf+  o)  Uf=  (  U<»  +  or  )/; 

i.e.  £/yalso  contains  f(x,y]  as  a  factor. 

*  Thus,  while  ^  =  c  is  readily  seen  to  he  the  equation  of  the  family  of  path-curves 
of  the  group  Uf=x       ~\~)'   '  ,   y—cx  —  o  is  another  form  for  it.    U(y  —  cx)=.  —  ex  -\-y 

docs  not  vanish  for  all  values  of  .i'  and;1;  but  it  does  vanish  for  those  values  satisfying 
the  i-quation  ot  the  path-curves  ;  sec;  (i-j)  bi-lo\v. 

f  It  is  presupposed  that  /(.v,y)  contains  no  repeated  (actors. 


§7  TIIKORY  OF  ONE-PARAMETER  r.ROUPS  ig 

In  the  same  way  it  can  be  shown  that  every  coefficient  in  (7)  con 
tains/^,  y)  as  a  factor,  whenever  6fdoes  ;  for  if 

u*f=  e(x,  y)f(x,  y),  u**f=  vv*/=  (  m  +  0o,)/. 

Hence  the  vanishing  of  Uf  whenever  f(x,  y)  docs  is  both  the  nccessaiy 
and  sufficient  condition  thatf(x,y)  =  Q  be  an  invariant  equation. 

In  case  Uf=  o  for  all  values  for  x  and  jy,  the  above  condition  is 
fulfilled.  But  this  we  recognize  as  the  condition  (9)  that  f(xt  y)  be 
an  invariant.  Hence,  not  only  is  f(x,  y)  =  o  a  path-curve,  but 
f(x,  y)  =  any  constant^  one  in  this  case. 

Remark.  —  It  should  be  noted  that 


may  vanish  because  £  =  o  and  r;  =  o  *  for  certain  values  of  the  vari 
ables.  In  general  these  two  equations  determine  a  finite  number  of 
values  of  the  variables.  Remembering  the  significance  of  £  and  rj, 
these  values  of  the  variables  are  left  unaltered  by  all  the  transforma 
tions  of  the  group  ;  so  that  the  points  having  these  values  for  coordi 
nates  are  invariant  points.  If  it  happens  that  £  and  rj  contain  a 
common  factor,  <»(x,y)t  it  is  obvious  that  u>(xty)  =  o  is  an  invariant 
curve,  in  that  every  point  of  it  is  invariant.  Following  Lie,  and 
desiring  to  preserve  the  significance  of  the  name,  we  shall  not  include 
this  class  of  invariant  curves  among  the  path-curves. 

Summing  up  the  results  of  this  and  the  preceding  section  we 
have  the 

THEOREM.  —  The  necessary  and  sufficient  condition  that  /Y.v,  v)  =  o 
be  invariant  under  the  group  Uf  is  that  Uf=  ofor  ail  values  of  x  and 
y  for  which  f(x,  y)  =  o,  /'/  being  presupposed  that  f(xt  y)  has  no 
repeated  factors. 

*  Still  another  possibility  is  that  J  =  o  and  *•    =  o  \vhrnrvor  f—  o.     But  this  is 

9*  dy 

excluded  by  the  restriction  thut/"(.i-,_y)  have  no  repeated  factors. 


2O  THEORY     OF   DIFFERENTIAL   EQUATIONS  §§  7,  8 

Points  whose  coordinates  satisfy  the  hvo  equations  £(x,  y)  =  o, 
77 (x,  jv)  =  o  are  invariant  under  the  group.  If  g(x,  y)  =  o  and 
rj(x,  y)  =  o  whenever  f(x,  y]  =  o,  ////>  CV/^T^  /V  composed  of  invariant 
points.  Curves  of  this  type  are  not  included  among  the  path-curves 
of  tJie  group. 

In  all  other  cases /(x,  y)  =  o  is  a  path- curve. 

If  Uf—  v  for  all  values  of  x  and  v,  f(x,  v)  is  an  invariant,  and 
f(x,  y)  =  any  constant  (including  zero)  is  a  path-curve. 

In  I,  £EEO,   rj=  i. 

.'.  n~x  =  i-onst.  is  the  equation  of  the  path-curves. 

There  are  no  invariant  points. 

In  II,  £  =  —  y,    f]  =  x. 

:.  n  —  x1  +jj'2  =  const,  is  the  equation  of  the  path-curves. 

There  are  no  other  invariant  curves.     The  point  x  =  o,  y  —  o  is  invariant. 

In  III,  £EEO,   77=^. 

/.  u~x  =  const,  is  the  equation  of  the  path-curves. 

y  =  o  is  an  invariant  curve,  each  point  of  which  is  invariant. 

Ex.  Examine  for  invariant  curves  and  points  the  groups  appear 
ing  in  the  problems  of  §  4. 

8.  Invariant  Family  of  Curves.  —  A  family  of  curves  is  said  to  be 
invariant  under  a  group,  if  every  transformation  of  the  group  trans 
forms  each  curve  into  some  curve  of  the  family.  We  shall  consider 
at  this  time  families  containing  a  single  infinity  of  curves  only,  that  is, 
those  whose  equations  involve  a  single  parameter  or  arbitrary  constant. 
Writing  the  equation  of  the  family  in  the  form 

/(*,  y)  =  t, 
it  will  be  invariant,  if 

/(*i, .>'i)  =/!>(*, y>  *\  «K*>  y>  0]  =  <*>(•*> y,  0  = f> 

is  the  equation  of  the  same  family  of  curves  for  every  value  of  /,  c 
and  c'  being  arbitrary  constants. 

A  single  infinity  of  curves  determined  by  an  equation  involving 
an  arbitrary  constant  is  equally  determined  by  a  unique  differential 


THEORY   OF   ONE-PARAMETER   GROUPS 


21 


equation  of  the  first  order,  of  which  the  equation  involving  the  arbi 
trary  constant  is  the  general  solution.  lff(x,'y)  =  c  and  <o(jc,y,  t)=c' 
are  to  be  the  same  family  of  curves,  these  equations  must  be  solutions 
of  the  same  differential  equation  of  the  first  order.  Hence  the  left- 
hand  member  of  the  one  must  be  a  function  of  that  of  the  ether,*  i.e. 

w  =  F(f). 
Making  use  of  the  relation  (7)  §  5,  viz. 


we  see  that/^,  }\)  will  be  a  function  of  f(x,  y)  for  all  values  of  /  if 
and  only  if  each  coefficient  in  the  expansion  on  the  right  is  a  function 
of/(jc,  _)').  In  particular  we  must  have 

(13)  Uf=F(f). 

If  (13)  is  true, 

Tfl-f TTfTf        TTJ?(  /\       ~~_\*K)  TTf       ~*_\Jl   J?(  f\ 

Uj  =  UUJ  =  Ur(T  )= —       —  Uj  =  —       —S(j), 

df  df 

which  is  again  a  function  of/ 

In  the  same  way  each  coefficient  on  the  right  is  seen  to  be  a  func 
tion  of/;  for  if  £/"/=<&(/),  U"+lf=  UUnf= 


Hence  (13)  is  both   the  necessary  and  sufficient  condition  that  the 
family  of  curves  /•/       \ 

be  invariant. 

*  The  differential  equations  arising  from  these  equations  are 


In  order  that  these  be  one  and  the  same  equation  it  is  necessary  and  sufficient  that 

d/ 
By 
dw 
dy 

But  this  is  the  condition  that  w  be  a  function  of/.     See  El.  />.>?'.  /•.</.,  Note  1  of  the 
Appendix. 


22  THEORY   OF   DIFFERENTIAL   EQUATIONS  §8 

Rejnark.  —  A  special  case  should  be  noted.  If  Uf  =  o  for  all  values 
of  x  and  yt/(x,  _)')—  c  is  a  family  of  path-curves,  each  one  of  which 
is  invariant,  hence  the  family  is.  This  particular  family  is  charac 
terized  by  the  fact  that  its  differential  equation  is 


The  problem  of  finding  all  the  families  of  curves  invariant  under 
a  given  group  Uf  will  be  considered  later  in  another  form  (§  18). 
The  general  type  of  such  families*  may  be  found  by  noting  that 
f(x,y)  must  satisfy  (13),  where  P(f)  is  some  function  of/,  not  de 
termined.  As  a  matter  of  fact,  F(f)  may  be  taken  as  any  convenient 
function  of/,  as  may  be  seen  from  the  following  consideration  : 

The    family  of  curves  f(x,  y)=  c   may    equally  well   be    written 
<£[/(.*:,  j)]=  const.,  where  <£(/)  is  any  holomorphic  function  of/. 
Applying  (13) 

«*(/)*=  cy-=*C/). 

This  will  be  any  desired  function  of/  say  O(/),  if 


Since  the  family  of  path-curves  is  excluded,  F(f)^.  o.  Hence  the 
function  4>  can  be  obtained  by  a  quadrature,  such  that  when  the  equa 
tion  of  the  invariant  family  of  curves  is  written  4>[/(,r,  y]\  =  const. 
the  right-hand  member  of  (13)  will  assume  the  desired  form  O(/). 

In  the  case  of  I,  equation  (13)  is  Uf=  &  =  F(f}. 

dy 

From  the  corresponding  system  of  ordinary  differential  equations 

<tx_d£_     df 
01        /?(/) 

*  In  this  discussion  the  family  of  path-curves  is  excluded,  since  a  method  for  find 
ing  these  curves  has  already  been  given  ($  7). 


§§8,9  THEORY   OF   ONE-PARAMETER  GROUPS  23 

the  general  solution  is  seen  to  be  of  the  form 


where  ^  is  an  arbitrary  function,  and  <£  =  f  —  •£—      Solving  for  /,  this  takes  the 

J  *\fj 

form  f=$(y  —  ^(x}}. 

The  most  general  family  of  curves  invariant  under  the  group  Uf=  =*•  is  then 

dy 

*(y  —  lK-*))  =  const.,  or  simply  y  —  ^(x)  =  c. 

Geometrically  this  is  obvious  at  once.  For  such  an  equation  represents  a 
family  of  curves  all  of  which  may  be  obtained  by  moving  any  one  of  them  con 
tinually,  in  either  direction,  parallel  to  the  axis  of  y. 

In  II,  —  y^f-  -\-x  §f-=  F(f}  leads  to  —  =  ^-'  =  —  ^—  ,  whence  the  general 
d-x         dy  —yx       /'(/) 

solution  is  of  the  form  tan'1  ¥.  —  0(/)  = 
or  /  «  *  (tan-1  £- 

The  equation  ~v  =  t,  representing  the  family  of  straight  lines  through  the  origin 

x 
is  a  simple  example  under  this  head,  as  is  immediately  obvious  geometrically. 

As  an  exercise,  the  student  may  show  that 


is  a  general  type  for  III,  while        x^\  ~  ]  —  c 
is  such  for  IV.     Simple  examples  are 


A--  +  2-  =  I,  a  family  of  central  conies  of  fixed  transverse  axis  for  III, 

<t.\-~  +  |8v2  =  r,  a.  family  of  similar  central  conies  for  IV, 
as  is  readily  obvious  geometrically,  and  as  may  be  verified  easily  analytically. 

9.  Change  of  Variables.  —  The  form  of  the  transformations  of  a 
group  depends  upon  the  choice  of  variables  that  are  operated  upon 
by  them. 

Thus  it  is  obvious  that  while  the  group  of  rotations  II   affecting  the  rectangu 
lar  coordinates  is 

.TI  =r  x  cos  a  —  y  sin  a,  y\  =  x  sm  a  +  y  cos  a, 


24  THEORY   OF   DIFFERENTIAL    EQUATIONS  §9 

when  operating  upon  polar  coordinates,  it  is 

/>i  =p,    Oi  =  0  +  a, 
which,  in  form,  is  identical  with  the  group  of  translations  I. 

To  find  the  effect  of  the  change  of  variables  * 
(14)  x=F(x,y),    y=*(x,y)9 

which,  of  course,  carries  with  it 


on  the  form  of  the  finite  transformations  of  the  group 
(i)  xl  =  <j>(x,  y,  a),  yL  =  \f/(x,  y,  a\ 

x,  y,  xl}  \'i  must  be  eliminated  from  the  six  relations,  (14),  (14'),  (i) 
and  the  resulting  two  relations  solved  for  xl  and  y^  This  elimination 
is  usually  effected  by  solving  (14)  and  (14')  forx,y,  xl}  ylt  and  substitu 
ting  these  in  (i). 

*The   introduction  of  new  variables  in   a  transformation  involves  the   following 
processes  : 

Designating  by  S  the  transformation  of  variables  (14),  or  (14').  a°d  by  5""1  its  inverse 

x  =  F(x,  y),  y  = 


obtained  by  solving  (14)  for  x  and  y,  the  new  coordinates  (x,  y)  of  any  point  are  ex 
pressed  by  means  of  S~l  in  terms  of  the  old  coordinates  (x,  y).  These  in  turn  are 
transformed  by  (i)  or  Ta  ($  i)  into  (x1,  y^)  of  the  new  point.  Finally  S  transforms  the 
latter  into  (x\,  y\),  the  new  coordinates  of  this  point.  Designating  by  Ta  the  transfor 
mation  in  the  new  variables  corresponding  to  Ta  in  the  old,  the  above  may  be  expressed 
symbolically 


The  transformation  Ta  is  known  as  the  transform  of  Ta  by  S. 

That  the  aggregate  of  the  transforms  of  all  the  transformations  of  the  group  (i)  form 
a  group  follows,  of  course,  from  the  fact  that  the  transformations  imply  certain  opera 
tions  which  are  independent  (except  as  to  form,  but  not  as  to  effect)  of  the  kind  of 
variables  operated  upon  by  them.  It  is  very  easy  to  verify  this,  however,  as  follows: 


TaTb  =  S-i  /a.svf-i  TbS  =  S-i  7  a  TbS  =  S--1  TeS  =  Tc 
since  6'A'-1  is  the  identical  trunslormation,  and  7'0  /i>  —  Te  (§  i). 


§9  THEORY   OF  ONE-PARAMETER   GROUPS  25 

In  the  case  of  the  above  example  the  formulae  for  the  change  of  variables  will 

be  chosen  in  the  inverse  form 

x  —  p  cos  6,  }'  =  p  sin  0. 

Eliminating  x,y,  x\,y\, 

Pi  cos  0i  =  p  cos  6  cos  a  —  p  sin  6  sin  a  =  p  cos  (0  +  #)» 
pi  sin  0i  =  p  cos  6  sin  #  +  p  sin  0  cos  a  =  p  sin  (0  -f-  #). 

Whence,  solving  for  p\  and  0i, 

pi  =  P,   0i  =  0  +  «. 

(The  other  possible  solution,  pi  =  —  p,  0i  =  0  -f-  IT  +a,  while  exactly  the  same 
geometrically  is  not  to  be  used  here,  since  the  above  transformation  must  reduce 
to  the  identical  one  for  a  =  o.  In  the  above  transformation  of  variables,  it  is 
understood  that  p  =  +  W2  -f-/2)- 

M 

In  general,  the  actual  work  required  to  carry  out  this  process  is 
long,  to  say  the  least  ;  on  the  other  hand,  the  problem  of  finding  the 
new  form  of  the  infinitesimal  transformation  is  a  very  simple  one. 
For,  remembering  that 


a* 


Similarly  r\(x,  y)  =  Uy. 

Hence 


,t 
dx  dy 

where  Ux  and  Uy  are  to  be  expressed  in  terms  of  x  and  y  by  means 
of  (14). 


26  THEORY   OK   DIFFKRF.NTIAL    EQUATIONS  §§  9,  10 

In  the  above  example,  choose  (14)  in  the  form 

p  = 
Since  u= 


10.  Canonical  Form  and  Variables.  —  It  is  always  possible  theo 
retically,  and  often  practically,  to  find  the  change  of  variables  that 
reduces  the  group  to  a  desired  form.  Thus,  in  order  to  have  the 
group  take  the  form 


any  convenient  pair  of  independent  solutions  of 


=^  y), 


may  be  taken  as  the  new  variables  x  and  y.     In  particular,  to  reduce 
the  group  to  one  of  translations  in  the  direction  of  the  axis  of  y, 

when  it  takes  the  form  U/=  -<-,  the  equations  to  be  integrated  are 


061) 


The  first  of  these  is  (9),  §  6  ;  so  that  for  x  may  be  taken  any  con- 
vc'nient  invariant  of  the  group,  //(x,y). 


§  io  THEORY   OF  ONE-PARAMETER   GROUPS  27 

To  solve  the  second  equation,  I/igrange's  method  leads  to  the  sys 
tem  of  ordinary  differential  equations 

dx  __  dy  _  dy 

7  =  7~T' 

which  are  equations  (5),  §  4.     Making  use  of  the  fact  that  u (x,  y)  = 

const,  is  the  solution  of  —  =  ^~,  y  may  be  obtained  by  a  quadrature.* 

£        *7 
Following  Lie  we  shall  say  that  the  group  is  in  the  canonical  form 

when  it  has  the  form  Uf=  — ,  and  the  variables  which  reduce  it  to 

oy 

this  form  will  be  called  canonical  variables.  The  above  result  may 
then  be  stated  : 

Every  group  can  be  reduced  to  the  canonical  form  Uf^  —  *     In 

g  dy 

order  to  Jin d  the  canonical  variables,  it  is  only  necessary  to  solve  the 
differential  equation  of  tlie  first  order 

dx  _  dy 

*"*' 

and  to  follow  this  witJi  a  quadrature.  In  case  an  invariant  of  the 
group  (or  what  is  the  same  thing,  the  equation  of  its  path-curves)  is 
known,  a  quadrature  alone  is  necessary. 

Remark.  —  If  the  equations  (16)  cannot  be  solved  readily,  it  may 
be  practicable  to  find  the  canonical  variables  for  both  the  original 
and  the  desired  forms  of  the  group.  A  proper  combination  of  these 
will  then  give  the  required  transformation  of  variables. 

In    II,  £=  —  y,  TJEE.T.     Here,  as  was  seen  (§  4),  WEE.T-  +  J'-.    v      tan""1*^- 

x 

These  ate  a  possible  set  of  canonical   variables.      I  hit   it   is   customary  to  choose 

V//  instead  of  u  for  X,  thus  giving  the  usual  polar  coordinates.     In   III,   £^O, 

*  Inspection  of  equations  (6),  ^  4  slioxvs  that  the  transformation  X  "(•*•', y),y  = 
v(x,y)  reduce.-,  tiic  i;roup  to  lli-:  torin 

I  xi  -  x,  j/i  =  y  4- 1. 


ft 

28  THEORY   OF    DIFFERENTIAL   EQUATIONS  §§  10,  n 

77  =  y.     Here,  as  was  also  seen  (§  4),  u  =  x,  v  =  \ogy.     In  IV,  —  —  -^-    gives 

x        y 

u=-*',    which    may  be    taken   as   x.     By  composition    the    system  of  equations 

dx      dy      dy     .         dx  -f  dy      du         , 

—  =  -*-  —  —  gives  -     £—  £  =  -3tj   whence  y  =  log  (x  +}'). 
x        y        I  x  -j-  y          I 

Another  set  of  canonical  variables  for  this  group  is  of  some  interest.     By  com 
position,  after  having  multiplied  numerator  and  denominator  of  the  first  member 

by  x  and  of  the  second  member  by  y,  we  have   x  x      ^  ?  —  —  ;   whence  y  = 

x-+*  I 


log  Vy2  +  y'1.     Choosing  this  form  for  y  and  tan"1  u  =  tan"1^  for  jr,  the  canoni- 

x 

cal  variables  are   very  similar  to   the   usual   polar   coordinates,  in   that  the  old 
variables,  in  terms  of  them,  are 

x  =  eU  cos  x,  y  —  eV  sin  x. 

From  their  nature,  it  is  obvious  that  in  passing  to  the  usual  polar  coordinates 
the  transformations  IV  assume  the  form  of  the  affine  transformations^II,  as  may 
also  be  verified  readily  analytically. 

Ex.  Find  the  canonical  variables  of  the  groups  in  the  problems 
of  §  4- 

11.  Groups  Involving  More  than  Two  Variables.  —  The  previous 
theory  of  one-parameter  groups  involving  two  variables  can  be  gen 
eralized  in  two  directions  :  the  number  of  variables  can  be  enlarged, 
and  the  number  of  parameters  can  be  increased.  In  this  section* 
will  be  considered  one-parameter  groups  involving  more  than  two 
variables  ;  and  as  the  argument  is  almost  the  same  for  n  variables  as  for 
three,  the  latter  number  will  usually  be  employed.  As  a  matter  of  fact, 
the  previous  arguments  for  two  variables  hold,  with  only  slight  modi 
fication,  for  a  larger  number  ;  hence,  as  a  rule,  only  the  facts  will  be 
given  here,  it  being  left  as  a  reviewing  exercise  for  the  student  to 
fill  in  the  supplementary  arguments. 

*  A  brief  extension  of  the  above  theory  to  groups  involving  more  than  one-pa 
rameter  will  be  given  in  Note  VI  of  the  Appendix. 


§ii  THEORY   OF  ONE-PARAMETER   GROUPS  29 

Starting  with  the  transformations 

('#!  =  <£(.*,  y,  z,  a), 
yi=*$(x,y,*>  <*), 
z\  =  \(x>}'>  z>  a\ 

where  </>,  ^,  x  are  supposed  to  be  generally  analytic,  independent, 
real  functions  of  x,  y,  z,  a,  they  will  constitute  a  Lie  group  provided 
the  set  has  the  following  properties  : 

i°  The  result  of  carrying  out  in  succession  two  transformations  of 
the  aggregate,  determined  by  any  two  values  a  and  b  of  the  parameter, 
is  the  same  as  performing  a  single  transformation  of  the  set  determined 
by  some  value  c  of  the  parameter,  where  c  is  a  function  of  a  and  b. 

2°  Solving  [i]  for  x,  y,  z  in  terms  of  xly  ylt  slt  the  resulting 
formulae  take  exactly  the  same  forms  as  [i],  some  function  of  a  tak 
ing  the  pfece  of  a.  In  other  words,  the  transformations  of  the  group 
occur  in  pairs  of  mutually  inverse  ones. 

As  a  consequence  the  group  contains  the  identical  transformation. 

A  group  of  this  type  contains  one  and  only  one  infinitesimal  trans 
formation  (§  2,  and  Remark,  Note  I  of  the  Appendix),  which  may  be 
written  * 
[3]  Uf=  t(x,  y,  »)      +  , (,,  y,  *)+{(*,  y,  s)^, 


where,  in  general, 

da      \da 

\          » 

ri"da=(daja: 
da      \da 

*  For  «  variables  we  have  likewise 


TIIKOKY  OF   DIFFERENTIAL   EQUATIONS 


The  finite  transformations  of  the  group  may  be  obtained  from  the 
infinitesimal  transformation  either  in  the  form  of  a  power  series  in  the 

parameter  (§  5)  /2 

xl=*x+  Uxf  +  U*x  —  +  •••, 

2  ! 
*!  ^  2  [ 

or  as  solutions  of  the  differential  equations  (§  4) 

r  n  dx^  d\\  <h,         _dt 

-"  £(      i     \  —  ~~7~~-     ~>      \  —  V(~~i)    '•"S  — — * 

If  ii\_(xi,  y\,  2j)  =  const,  and  /v^v,,  rh  z{)  =  const,  are  the  solutions  of 
the  first  two  equations  (not  involving  /),  and  r(.\'lt}\t  -/)  —  /=  const. 
is  a  third  solution  of  the  system  independent  of  the  other  two,  then 


determine  the  finite  transformations  of  the  group. 

In  both  these  cases  the  parameter  /  enters  in  such  a  way  that  /=  o 
gives  the  identical  transformation,  and  /=  —  /determines  the  inverse 
transformation.* 

*  In  the  case  of  n  variables,  the  development  form  of  the  finite  transformations  is 
exactly  the  same.  To  obtain  the  second  form,  the  system  of  differential  equations  is 


[Si 


and  their  solutions  are  of  the  form 


V 


^(.iV,  .v./,  ..-,  v,/)       =  Ul(.rlt  jro,  ••-,  xn), 


Primed  letters  are  used  here  to  designate  the  transformed  variables,  since  the  sub 
script,  previously  employed,  is  no  longer  available. 


§n  THEORY   OF  ONE-PARAMETER   (iKOLTS  31 

The  effect  of  a  finite  transformation  of  the  group  on  any  function 
/(x,y,  z),  is(§  5) 

[7]  /(*Wi,  sj  =/(*,>',  *)  +  Uft+  U*f~±  "- 

A  function  /(*,  y,  s)  is  invariant  under  the  group  Uf\i 


for  all  values  of  .r,  v,  s  (§  6). 

This  equation,  involving  three  independent  variables,  has  two  inde 
pendent  solutions.  Hence  a  one-parameter  group  in  three  variables 
has  two  independent  invariants.  Since  MI(X,  y,  z)  and  //o(.v,  v,  z)  are 
such  a  set,  every  invariant  of  the  group  is  a  function  of  //i  and  //2.* 

Those  points  whose  coordinates  satisfy  the  three  equations 

£(.v,  y,  z)  =  o,    rj(x,  y,  z)  =  o,    £(x,  y,  z)  =  o 

are  invariant  under  the  group  (§  7).  In  general,  that  is,  in  case  the 
three  functions  are  independent,  there  is  only  a  finite  number  of 
such  points.  But  if  only  two  of  the  functions  are  independent  (which 
will  show  itself  by  having  their  Jacobian  vanish,  without  all  of  its  first 
minors  doing  so)  the  two  independent  equations  will  be  the  equations 
of  a  curve,  every  point  of  which  is  invariant.  If  all  the  two-rowed 
determinants  in  the  Jacobian  vanish,  there  is  only  one  independent 
equation,  and  it  is  the  equation  of  a  surface,  every  point  of  which  is 
invariant  under  the  group. 

The  path-curves  are  obtained 

i°  either  by  eliminating  a  from  the  finite  transformations  of  the 
group  (§  i), 

2°  or  by  solving  the  system  of  ordinary  equations  (§  7) 

-i  '/v      t/       dz 


*  In  the  c«M  <>f  //  variables,  every  invariant  of  the  group  is  a  function  of  the  »—  i 
independent  ones  //,,  //.>,  •••,  un  j. 


32 


THEORY   OF   DIFFERENTIAL   EQUATIONS 


§u 


From  the  latter  we  see  that  if  //j  and  //.2  are  two  independent  invari 
ants  of  the  group,  //j  =  const,  and  u2  =  const,  are  the  equations  of  the 
path-curves. 

Each  of  the  surfaces  u\  —  const,  and  /A,  =  const,  is  invariant,  being 
made  up  of  an  infinity  of  path-curves  obtained  in  either  case  by  keep 
ing  one  of  the  constants  in  the  equations  of  the  path-curves  fixed  and 
allowing  the  other  to  run  through  its  full  range  of  values.* 

The  equation  f(xt  y,  z)  =  o,  or  the  surface  represented  by  it  is 
invariant  (§  7)  if 
[12]  Uf—  o  whenever/=  o,| 

provided  /  contains  no  repeated  factors.  (If  Uf  vanishes  because 
£  =  o,  11=0,  £=o  whenever  /=o,  every  point  of  the  surface  is 
invariant.) 

The  curve/j  (x,  y,  z)  =  o,/  (x,  y,  z)  =  o  is  invariant  if 

[12']          Ufa  =  o  and  £/2  =  o  whenever/  =  o  and/2  =  o, 

provided/  and/,  contain  no  repeated  factors  and  are  independent 
functions,  not  containing  a  common  factor.  This  last  condition 
assures  us  that  not  all  of  the  two-rowed  determinants  in  the  matrix 


dx 
df, 
dx 


By 
df2 
dy 


dz 
df, 
dz 


vanish  for  all  values  of  x,  y,  z. 


*  In  the  case  of  «  variables,  i°  holds  without  change  ;  in  2°  the  differential  equations 
of  the  path-curves  are 


£ 


£• 


and  their  finite  equations  are  «j  —const.,  //._,  •  <  (v/v/.,  •••  ,  //„  _i       const.,  where  //]//.,>,  •••  , 
n,,    i.  are  any»—  i  indej>endent  invariants.     l-!acli  <>i  t!u;   (//       i)  -way  spreads  in   n 
dinii-nsioMs  //!  =  const.,  u^  —  consf.,  •••  ,  ««-i  =  const,  is  invariant,  as  well  as  the  various 
spreads  of  lower  dimensions  obtained  by  taking  these  invariant  relations  two,  three, 
•  .  //    -i  together,  the  last  case  giving  the  path-curves. 
t  This  condition  holds  when  the  equation  involves  any  number  of  variables. 


§u  THEORY  OF  ONE-PARAMETER   ClROUPS  33 

The  argument  employed  in  establishing  this  theorem  for  a  curve  in 
three  dimensions  is  different  from  that  available  in  the  case  of  a 
surface  f(x,  v,  z)  =  o  (in  the  latter  case  the  one  employed  for  a 
curve  in  two  dimensions  (§  7)  applies). 

The  necessity  of  the  condition  is  seen  as  before  ;  for,  using 
formula  [7] 


H  f\(x\i  )\i  z\}  and  fi(x\,y\,  Zi)  are  to  vanish  whenever  /(jc,  j,  2) 
and  /*(x,y,  z)  do,  for  all  values  of  /,  it  is  necessary  that  f/l/=o 
and  £/2/=  °  whenever/!  =  o  and/,  =  o. 

The  sufficiency  of  the  condition  follows  at  once  from  the  fact  that 
since  *,  ,  -, 


all  along  the  curve  /1  =  o,/  =  o,  £,  77,  ^*  are  proportional  to  the 
direction  cosines  of  the  tangent  of  this  curve  at  each  point  (x,  y,  z)  ; 
that  is,  this  curve  is  the  path-curve  through  the  point  (x,  y,  z). 

Remark.  —  If  Vf\  =  o  whenever  ^=o,  and  Uf«  =  o  whenever 
/  =  o,  the  surfaces  /  =  o  and  /  =  o  are  separately  invariant  ;  and 
their  intersection  is  also  invariant.  In  the  case  under  consideration 
above,  however,  [12']  is  the  condition  for  invariance  of  the  curve 
without  regard  to  the  nature  of  these  surfaces. 

The  change  of  variables 

[14]  x  =  F(  >•,  y,  2),  y  =  ^.v,  v,  z\  z  =  vl/(.v,  r,  s) 

*  If  |  —  o,  17  =  o,  f  =  o  whenever/!  =  c,  /^  =  o,  every  point  on  this  curve  is  invari 
ant,  and  hence,  the  curve  itself  is  ;  so  that  the  sufficiency  is  also  established  in  this  case. 
But  such  a  curve  is  not  included  among  the  path-curves  of  the  group  (Remark,  §  7). 


34  THEORY   OF    DIFFERENTIAL    EQUATIONS  §  n 

• 

causes  the  infinitesimal  transformation  to  take  the  form  (§  9) 

[IS]      ^ff+^I^S- 

So  that  the  new  variables  satisfy  the  differential  equations  (§  10) 

&  =  &+*  +  &  =  &,,,:), 

ox         oy         oz 


In  particular,  when  |  =  o,  i]  =  o>£=il  the  group  is  said  to  be  in 
the  canonical  form*  If  the  equations  of  the  path-curves  are  known, 
the  canonical  variables  can  be  found  by  means  of  a  single  quadrature. 

To  illustrate  all  that  has  gone  before  consider  the  group  of  screw  motions 
x\  —  x  cos  t  —  y  sin  /, 
y\  —  x  sin  /  +  y  cos  /, 

21  =  2  +  ;///, 
where  m  is  any  constant. 

The  student  will  have  no  difficulty  in  proving  that  these  transformations  have 
the  group  property,  and  that  in  this  case  (§  i) 

Ttjtt=  Ttl+tt', 
also  l—  —  t,  and  /0  =  o. 

The  infinitesimal  transformation  is  readily  seen  to  be 


*Morc  generally,  the  group  will  !•<•  said  to  !x;  in  tin-  canonical  form  when  any  one  of 
£>  77»  4"  <'<J'«;ils  ;i  roustunt,  and  the  other  two  are  zero. 


§n  THEORY  OK  ONE-PARAMETER  r.uorps  35 

Conversely,  starting  with  the  infinitesimal  transformation  the  Unite  transforma 
tions  are  found  to  be,  using  [<SJ, 


(         t"1       t*  \          I         /3        /5  \ 

^xil  ---  (-  —  —  •  •  •    —  y\t  ---  1  ---  "•  1  =  *  cos  t  —  y  si 
\         2!       4!  J         \        3!       5!  j 

(t*  /."> 

/  --  -+     :—• 


sin  t. 

/2  /4 


_.  .     ,          ,  •  - , 

2!       4! 

z\  •=•  z  -\-  nit  =  z  -\-  nit ; 

r,  using  the  other  method, 


=  tan-1  Pi-  —  ^L  =  tan  -1  -'  —  —, 
,V|       m  x       m 


For  practical  purposes  it  will  he  simpler  to  replace  z,  in  the  second  equation 
hy  its  value  in  the  third  one.     Then 


tan-1^  =  tan"1-'1'  +  t, 
x\  x 

Zi  =  Z  +  ////. 

The  third  equation  is  already  in  proper  form. 

Ilr-  lusi  t\vo  equations  are  free  of  c,  and,  as  was  found  in  §  4,  reduce  to 
XL  =  .v  (-us  /  -  -  )'  sin  /, 
jj'!  —  x  sin  t  -\-  y  ci  >s  /, 

T\vo  iiKh'pendent  invariants  are  n\      .\-    '•    i':,   //  .     'tan    l!/       —•      Hence  th 
,  x      in 

pain-curves  are 

*2+;(2  =  r2f     ^-l^^i.^, 

X         III 

t—,  introducing  the  parameter  9, 
x  —  rcos/'.     r  =  rsin^,    z  =  »i(0  —  r\ 


lich  is  a  family  of  helices,  involving  the  arbitrary  constants  r  mid  r, 
If  m  ^  o  there  are  no  invariant  points. 


36  THEORY   OF   DIFFERENTIAL   EQUATIONS  §11 

Two  of  the  canonical  variables,  x  and  y,  must  satisfy  the  differential  equation 

_  ,df  +  xQf±.,n  Of-  o 
d*        dy         dz 

while  the  third,  z,  must  satisfy 

df  ,       df          df 

—  V  —  -f-  x  •—  4-  m  —  =  i. 

d*        dy         dz 

Knowing  the  invariants  of  the  group,  u\  and  u»,  we  may  put 

x  —  VV2  +  y2,    z/  =  tan'1  %  —  —  • 
x       m 

By  inspection,  z  may  take  the  simple  form 


Solving  for  the  old  variables,  the  formulae  of  transformation  of  variable0  »*e 
x  =  xcos  (y  +  z),  y  =  ysin  (y  +  z),    z  =  mz. 

It  is  obvious  that  the  change  to  cylindrical  coordinates 
x  =  p  cos  6,  y  =  p  sin  0,    z  =  z 

reduces  the  group  to  the  form 

Pi  —  Pi    6\  •=  0  -\-  t,    z\  =  z  +  w/"» 
which  is  a  group  of  translations,  but  not  in  the  canonical  form. 

Discuss  as  was  done  in  the  text  the  following  groups : 

"C*  Y          O  y*     /7  Y*          U     —  il  V          ?          --,   i* 

Ex.    3.    xl  =  rt.v,    T!  =  ay,    zl  =  az. 

Ex.    5.   xl  =  ea (x  cos  a  —y  sin  «),  yl  =  e°  (x  sin  a  +y  cos  «),  ^  =  eaz. 


CHAPTER    II 
DIFFERENTIAL   EQUATIONS   OF  THE  FIRST  ORDER 

12.    Integrating  Factor.  —  We  have  seen  (§  8)  that  if  <j>(x,y) 
const,  is  a  family  of  curves  invariant  under  the  group 


(13)  U* 

Moreover,  it  was  also  shown  in  §  8  that  if  the  curves  of  the  family 
are  not  path-curves  of  the  group,  the  equation  of  the  family  can  be 
chosen  in  such  form  that  the  right-hand  member  of  (13)  shall  be 
come  any  desired  function  of  (f>.  In  particular,  there  is  no  loss 
in  assuming  the  equation  so  chosen  that  this  right-hand  member  is 
i  •  for  if  a  given  choice  <£  =  const,  leads  to  F(<$>\  the  selection 

<$(<£)  =  const.,  where  <$(<£)  =   I     l  ***   ,  will  give  £/<£>(<£)  =  i. 

J  F(<f>) 

Suppose  now  that 
(17)  Mdx  -\-Ndy-Q 

is  a  differential  equation  whose  family  of  integral  curves 
(  1  8)  $(x  ,  y)  =  const. 

is  invariant  under  the  group  Uf,  the  integral  curves  not  being  path- 
curves  of  the  latter.  Let  <£  be  so  chosen  that 

(19)  ^  =  ««*+,|4=I. 

dx         By 

37 


38  THEORY   OF   DIFFERENTIAL   EQUATIONS  §  12 

Since  (18)  is  the  solution  of  (17), 

,.      d<f>   ,     .   dd>   , 
d<\>  =  -^-dx  +  --?-dy  =  Q 
ox  ay 

must  be  the  same  equation  as  (17)  ;  hence 

0<j>      d$ 
dx      dy 


(20) 


- 

dx  dy 


From  equations  (19)  and  (20)  the  values  of  -—  and  —  are  found 

ox  dy 

to  be 


Mdx 


Hence  the 

THEOREM.*  —  If  the  family  of  integral  curves  of  the  differential  equa 

tion  Mdx  +  Ndy  =  o  is  left  unaltered  />v  the  youp  Uf~^-  +  »4^i 

Av         dy 

is  an  integrating  factor  of  the  differential  equation. 


Remark  i.  —  This  theorem  ceases  to  hold  in  case  the  curves  (18) 
are    path-curves   of  the    group    Uf.      In    this   case    (19)    becomes 

£—+77  —  =  o;    whence,  taking  account    of  (20),  £Af+-nN=o. 
dx         dy 

As  a  matter  of  fact,  it  is  obvious  that  in  this  case  the  curves  (18), 
being  the  integral  curves  of  (17),  are  the  path-curves  for  every  group 
of  the  type 


*  This  theorem  of  Lie  was  first  published  l>y  him  in  the  Vcrhandlungen  der  Gi-sell- 
scliaft  dcr  XN'isscnhchaftun  zu  Christiania,  November,  1874. 


§12        DIFFERENTIAL    EQUATIONS   OF  THE   FIRST   ORDER          39 

where  p(x,  y)  is  any  holomorphic  function  of  x  and  y.  Such  groups 
are  said  to  be  trivial  for  purposes  of  assisting  in  solving  the  differen 
tial  equation  (17). 

Remark  2.  —  At  times  it  is  obvious  from  the  nature  of  the  problem 
that  the  family  of  integral  curves  is  invariant  under  a  certain  group. 
This  will  be  found  to  be  the  case  in  the  following  examples  : 

Ex.  1.  Find  the  curves  whose  tangent  at  each  point  makes  an 
isosceles  triangle  with  the  axis  of  x  and  the  radius  vector  to  the  point 
of  contact. 

This  family  of  curves  is  clearly  invariant  under  the  similitudinous  group 
£//=.*•—  -f  yQ£-  Its  differential  equation  is 

dx    *  dy 

dx      x  dy  I  dv  \ 2  ,         dy          

!  +Z£L~~ </x'       \<tx)  '      <t*~ 

x  dx 

Reducing  to  the  form  (17),  which  is  characterized  by  being  of  the  first  degree 
in  dx  and  dy, 


(x  ±  V**  +  jP)  dx  +  y  dy  —  o. 
The  integrating  factor 

I  I 


+ 


Integrating,  log  (x±  V-r'-^+y2)  =  const,   or  x  ±  V-rM 

This  reduces  at  once  to  v2  =  c-  —  2  r.r,  a  family  of  parabolas  having  the  origin 
as  common  focus  and  the  axis  of  .r  as  common  axis. 

Ex.  2.  Find  the  curves  such  that  the  radius  vector  to  each  point 
makes  an  isosceles  triangle  with  the  tangent  at  the  point  and  the 
axis  of  x. 


4<D  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§  12,  13 

Ex.  3.  Find  the  curves  such  that  the  length  of  the  radius  vector 
to  each  point  equals  the  tangent  of  the  angle  between  the  radius 
vector  and  the  tangent  to  the  curve  at  that  point. 

Ex.  4.  Find  the  curves  such  that  the  radius  vector  to  each  point 
makes  a  constant  angle  with  the  tangent  to  the  curve  at  that  point. 

Ex.  5.  Find  the  curves  such  that  the  perpendicular  distance 
from  the  origin  to  the  tangent  to  a  curve  at  any  point  is  equal  to 
the  abscissa  of  that  point. 

13.  Differential  Equation  Invariant  under  Extended  Group.  — 
While  at  times  it  is  possible  to  tell  from  the  nature  of  the  problem 
whether  the  integral  curves  of  a  differential  equation  form  an  invariant 
family  under  a  certain  group,  it  is  desirable  in  order  to  extend  the 
usefulness  of  the  theorem  of  the  previous  section,  to  be  able  to  tell 
when  this  is  the  case  from  the  form  of  the  differential  equation  itself. 

A  point  transformation 


carries  with  it  the  transformation 

cty 
d\,       dx  dv 


dx  dy 

tt+tty 

.        dx        rV  ., 

*"-§73£"3^#- 

dx      dy 

*  This  is  called  a  point  transformation  because  it  transforms  the  point  (x,y)  into 
(  r,.  r, ).  It  thus  transforms  the  various  points  of  a  curve  F(x,y)  =  o  into  the  corre 
sponding  points  of  some  other  curve  /'\(-*'i,>'i)  =  o,  and  may  therefore  be  said  to  trans 
form  the  curve  F(x,y)  =  o  into  ^(x^y^  —  o. 


§  13        DIFFERENTIAL  EQUATIONS   OF  THE   FIRST   ORDER          41 

where  v'  =  —  and  r/  =  —  •     Since  ^  is  a  function  of  x,  y,  v'  only,  it 
dx  dx^ 

follows  that  the  point  transformation  implies  the  transformation 

fy   ,  fy 

~ 


,        vt 

"•T"    i  —  a"~«' 

djc      ay 

affecting  the  three  variables  x,  y,  y'.  The  latter  transformation  is 
known  as  an  extended  point  transformation* 

Starting  with  the  one-parameter  group  of  point  transformations 

(i)  .T!  =  4>(x,  y,  a),  )\  =  $(x,y,  a) 

it  is  easily  seen  that  the  corresponding  extended  transformations 

(21)         xv=  <j>(x,y,  a),yi  =  t(x,y,  *)iJfc'-  =  ^-3jit(#,j».7/«) 

also  constitute  a  one-parameter  group  in  the  three  variables  xt  y,  y'. 
For,  since  the  equations  of  a  point  transformation  are  precisely  the 
first  two  of  the  corresponding  extended  transformation,  and  since  the 
third  equation  of  the  latter  is  determined  uniquely  by  the  first  two, 
the  fact  that  the  transformations  (i)  have  the  group  property  (§  i) 
predicates  the  existence  of  the  group  property  in  the  case  of  (21). 

Thus  if  a  and  /;  are  any  two  selected  values  of  the  parameter,  the  result  of  per 
forming  successively  the  two  point  transformations 

*i  =  0O,  y,  «),    y\  —  <AO,  ;',  «) 

and  x«  =  0(>i,  _>'!,  /;),  y»  =  \f/(xi,  yi,  b) 

is  *2  =  «/>(-*-,  y,  Ot    yt  =  $(*>  y>  0 

*  An  extended  point  transformation  is  a  special  kind  of  a  contact  transformation 
($  49)  ;  for  it  transforms  (.v,  v,  /)  into  (.»•],;',,  ;•,'),  where,  if  (.v,j)  is  some  point  on  some 
curve  /''(-ft  y)  =  o,  y'  is  tin-  .slopr  of  tlir  t.  indent  to  tin-  curve  at  th.it  point  and  1^'  is  the 
slope  of  curve  /^(.Vj,  y\)  =  o  (into  which  the  other  is  transformed  by  thr  point  trans 
formation)  at  the  corresponding  point  (-»"],  /j).  Since  the  value  of  _V]'  depends  upon 
x,y,y'  only,  any  curve  tangent  to  F(x,  v)  =oat  (.v,y)  will  be  transformed  into  a  curve 
tangent  to  f>\  (-«'i,>'i)  =  o  at  the  point  (.Vj,  _vj). 


42  THEORY  OF   DIFFERENTIAL   EQUATIONS  §  13 

where  c  is  a  function  of  a  and  b.     This  follows  from  the  group  property  of  (i). 
In  the  case  of  the  corresponding  extended  transformations 


.  ,t 

"0  (x,y,  a 
and       x. 


the  result  of  replacing  .TI  and  j'i  in  the  first  two  equations  of  the  second  trans 
formation  by  the  values  given  in  the  first  transformation  is  therefore 


Hence  in  the  last  equation  of  the  second  transformation, 


In  exactly  the  same  way,  the  fact  that  a  value  of  the  parameter 
exists  giving  the  identical  transformation  for  the  group  (i),  and  also 
the  fact  that  the  transformations  of  (i)  can  be  separated  into  pairs 
of  mutually  inverse  transformations,  assure  these  same  properties  for 
the  transformations  of  (21).  The  latter  therefore  constitute  a  Lie 
one-parameter  group.  This  group  is  known  as  the  once-extended 
group  corresponding  to  (  i  )  . 

With  Lie,  we  shall  write  as  the  symbol  of  the  infinitesimal  trans 
formation  of  the  once-extended  group 


(22) 


where,  as  before,   t  =  ^,  ^=  -,  while  r,'  =  ^  =&,( 


It  was  seen  in  §  4  that,  with  a  proper  selection  of  the  parameter, 
,  =    (          ,  ami,  for  any  function/,       = 


§13         DIM'KRKNTIAI.    F.OUAT1OXS    OF   Till';    M  RST   ORDER  43 

In  a  sense  then  8  is  a  differential  operator,  so  that  8  and  d  are 
commutative  operatois  ;  thus,  for  example, 


,       8    dy\      8<r  '          -  &*v  V 

Hence  r?  =  ^       *    ,  =  — -, 7-7 — -^ —  —  — j ,   — -: 

'      badx         tfa  (ifcr  dx        dx    dx 


dx     J  dx 


Remark,  —  Attention  should  be  called  to  the  fact  that,  while  /  is 

ual  to   -^,  -n   is  usually  diff 
dx 

hand  member  of  (23),  we  have 


equal  to   -^,  -n   is  usually  different  from  -£•     Expanding  the  right 
dx  dx 


where  it  is  to  be  noted  that  rf  is  a  quadratic  polynomial  in  7'  when 


Given  a  differential  equation  of  the  first  order 
(25)  /(*,*/>*»o, 

the  effect  of  any  transformation  (i)  on  the  variables  .v  and  v  is  to 
transform  the  differential  efjuation  (considered  as  an  equation  in  the 
three  variables  .v,  r,  /  )  by  the  corresponding  extended  transformation 
(21).  The  family  of  integral  curves  of  (25)  is  invariant  under  the 
group  if  each  integral  curve?  is  transformed  into  some  curve  of  the 
family  by  every  transformation  (i).  Hence  every  transformation  (21) 


44  THEORY  OF   DIFFERENTIAL   EQUATIONS  §§  13,  14 

must  leave  the  differential  equation  unaltered.     The  condition  for 

this  is  ([12],  §  n) 


(26)        ir/~  £      +  j      +  rj'       =  o  whenever/(>,  jr,y)=  o. 

Hence  the 

THEOREM.  —  The  family  of  integral  cuives  of  the  differential  equa 
tion  f(x,  y,  y')  =  o,  and,  therefore,  the  differential  equation  itself,  is 
invariant  under  the  group  Uf  if  U'f=  o  whenever  f=  o. 

In  the  case  of  II,  £=—  y,  TJ  =  X.  Hence,  from  (23)  r?'  =  i  +?'"'  The  ex 
tended  group  of  rotations  is  then 

U^.SfS^df 
'  d*        dy  d/ 

The  differential  equation  of  the  family  of  lines  <-  =  c  (which  is  invariant  under 
II)  is  xy'  —  y  =  o.  Here 

Vl(xyi-y)  =  -yy»  -  *  +  (l  +/*)*  =/(^/-j). 
This  vanishes  whenever  xy'  —  y  does. 

14.  Alternant.  —  Let  U\  and  60  be  any  two  homogeneous  linear 
partial  differential  operators  * 

^i  =  li(^,  .)')—  4-  »7i(.v,  y)—  , 

C/2  =  &(#,  >')—  +  ^(.v,  7)—  . 
Then 


(27)    •••  i^J 

*  Yor  the  sake  of  simplicity  \v<>  shall  suppose  that  two  variables  are  involved.     But 
tiiis  entire  section  hoids  without  any  inodilication  tor  -v  variables. 


§§I4,  15      DIFFERENTIAL  EQUATIONS  OF  THE   FIRST  ORDER       45 

Writing  l\U,f-  UMJ=(yjJ?)f* 

the  operator  (U\U^),  which  is  known  as  the  alter  nan  t\  of  i/\  and  £/,, 
is  seen  to  he  one  of  the  same  type  as  U{  and  U>. 

The  following  properties  of  alternants  are  immediate  : 


i,  U,±  tf8)  = 


]  5.  Another  Criterion  for  Invariance  of  a  Differential  Equation 
under  a  Group.  —  A  second  form  for  expressing  the  condition  that  a 
group  leave  a  differential  equation  unaltered  plays  a  very  important 
role  in  the  further  development  of  the  theory.  It  was  seen  (§  1 2), 
that  if 
( 1 8)  <£  (x,  _y)  =  const. 

is  the  solution  of 

<£  is  a  solution  of  the  partial  differential  equation  (20) 

,7dcf>       jij-dtb 

( 2  o )  A<p  =  IV  — —  —  J.V1  — -  =  O. 

dx  By 

Moreover,  if  the  family  of  curves  (i  8)  is  invariant  under  the  group  Uf 
(without  being  path-curves  of  the  latter),  <£  may  be  so  chosen  that 

Consider  now  the  alternant  of  f/and  A  (§  14) 

Af  Af 

(37) 


*  Lie  writes  (  U±  U.,}  or  (  I  \f,  U.,J}  instead  of  (  l\  ('.,)  f. 
f  Also  sometimes  called  the  commutator  of  f  \  and   /  j. 


46  THKORY   OK    DIKKKRKNTIAL   EQUATIONS  §15 

Because  of  (28)  and  (19)     (UA)$  =  U(o)—A(i)=o. 
(29)  A  (UN-A^-WM+A^*^. 

Since  cf>  is  a  function  of  at  least  one  of  the  variables  x  and    r, 

-£  and  —  are  not  both  identically  zero.     Hence  the  coefficients  of 
dx  dy 

(29)  must  be  proportional  to  those  of  (28);  i.e. 
(3o) 


or  UN-A£  =  *N,     UM+A-n 

Putting  these  in  (27) 

(31)  (UA)f=\(x,y)Af.. 

Hence  (31)  is  a  necessary  condition  that  the  integral  curves  of  (17) 
be  invariant  under  Uf. 
Conversely,  if  (31)  holds 


because  of  (28).     Hence  AU<f>  =  o. 

Since  every  solution  of  (28)  is  a  function  of  <£ 


This  is  the  condition  [§  8,  (13)]  that  the  family  (18)  be    invariant 
under  the  group  Uf.     Hence  the 

THEOREM.  —  The  necessary  and  sufficient  condition  that  the  differen 
tial  ('({nation  M  dx  +  N  dy  =  o  be  invariant  under  the  group  Uf  is 

(31)  (UA)f=\(X,y)Af 

Af=N^  -M-£. 
dx  dy 

*  The  common  ratio  \(x,y)  is,  at  most,  a  function  of  the  variables.     It  may  be  a 
constant  or  zero. 


§  15        DIFFERENTIAL   EQUATIONS  OF  THE   FIRST  ORDER         47 

The  condition  (31)  was  found  independently  of  what  has  gone  before.  It 
may  be  obtained  at  once  by  means  of  (26).  It  is  suggested  as  an  exercise,  that 
the  student  do  this.  Here  f(x,  y,  /)  E=  Af+  Ny1  .  The  expanded  form  of  ij', 
given  by  (24),  must  be  employed. 

This  theorem  leads  to  another  one,  of  some  interest,  which  is,  as 
a  matter  of  fact,  the  converse  of  the  theorem  of  §  12. 
If  £(JT,  j)  and  t]  (x,  y)  are  any  two  functions  such  that 

i 


is  an  integrating  factor  of 

(17)  M  dx  -}-  N  (fy  =  o, 

d(       N      \      d(       M      \ 


or 

dx  dx  dx  dx  dy  dy  dy 


dy 
Dividing  by  J/7Vand  rearranging  the  terms, 


dx  dy  dx  dy  __     dx  dy  dx  dy 

"  N  M 

UN-Ai      UM+Ai) 

(30)  -JT       ~M~ 

from  which  follows  (31)  as  before.  Hence,  if  /JL(X,  v)  is  an  integrat 
ing  factor  of  tJic  differential  equation  M  dx  +  N  dy  =  o,  and  ±(x,y) 
and  rj(x,  y)  are  any  holonwrphic  functions  of  tlic  variables  satisfying 
the  relation 


*  See  EL  Dif.  Eq.  \  7. 


48  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§  15,  16 

the  differential  equation  is  invariant  under  the  group 

w£  +  ,3£. 

dx         dy 

Since  £  and  77  are  subject  to  the  single  condition  (32),  one  of  them  may 
be  chosen  at  pleasure,  and  then  the  other  is  determined  uniquely. 
Hence,  starting  with  an  integrating  factor  of  a  differentia!  equation 
of  the  first  order,  an  infinite  number  of  groups  can  be  found  which 
leave  the  differential  equation  unaltered. 

It  will  be  seen  in  §  17  that  the  general  expression  for  such  groups  involves 
two  arbitrary  functions.  As  a  matter  of  fact,  this  can  also  be  seen  from  the  form 
of  (32).  For  if  /j.  is  an  integrating  factor  giving  f»(Aft/jr  +  Ndy}  =  du,  then  for 
/•'(#)  any  function  of  it,  /"/•'(«)  is  also  an  integrating  factor.  (See  El.  Dif.  Eq. 
§  5.)  Using  this  as  the  right-hand  member  of  (32),  and  selecting  £(>,  jj/)  arbi 
trarily,  77  =  —  ^  '/^  .  The  general  type  of  group  leaving  (17) 


unaltered  may,  therefore,  be  put  in  the  form 

T7J- >-   f  ..        ..\      Of      ,        I  I 


where  £  and  /'are  arbitrary  functions. 

16.  Two  Integrating  Factors.  —  Since  the  knowledge  of  a  group 
which  leaves  a  differential  equation  unaltered  gives  an  integrating 
factor,  thus  reducing  the  problem  of  solving  the  differential  equation 
to  a  mere  quadrature,  it  should  be  expected  that  the  knowledge  of  a 
second  group  which  leads  to  a  distinct  integrating  factor  still  further 
simplifies  the  problem  of  solving  the  equation.  This  is  actually  the 
case. 

Suppose  /M!  and  /xo  to  be  two  integrating  factors  of  (17).     Then 


dy  r).v 


dy       dx      Ml (      'dx  '      "  dy)     /xA" '  dx       "  dy 


§§  i6,  17    DIFFERENTIAL   EQUATIONS   OF  THE  FIRST  ORDER      49 

i  dii      d(log  u)       i  3u      3(log  u) 

Remembering  that   --C  =  -3   pLC/,         -g  —    v    fe^?  and 
|u,  o.r          •  ojc          /x  qj  oy 

log  /xx  —  log  /x2  =  log  ^,  the  last  equation  becomes 

/x, 

AT  3'  /,      MI\       1^-3  /,      uA 
3jcV  )""      F  V  )  =  °  ' 


/>.,  log  —  is  an  integral  of 
(28)  A/  = 


- 

dx  dy 


Hence  ^  is  also  an  integral  of  (28),  and 

^2 

^  =  const. 

P-2 

is  a  solution  of  (17).  So  that  the  knowledge  of  two  integrating  factors 
gives  the  solution  of  the  differential  equation  without  any  analytic 
work  whatever. 

Remark.  —  It  is  interesting  to  note  that  in  the  proof  usually  given 
for  the  theorem  that  when  one  integrating  factor  /a  is  known,  an  in 
finite  number  of  others  can  be  found  [viz.  if  p.(Mdx-\-  Ndy)=:duy 
then  pF(u)  is  an  integrating  factor  where  F(u)  is  any  function  of  //],* 
all  possible  integrating  factors  are  found. 

17.  General  Expression  for  Group  under  which  a  Differential  Equa 
tion  is  Invariant.  —  We  have  just  seen  that  if  U^f  and  £/,/are  any 
two  groups  which  leave  the  equation  (17)  unaltered, 


P., 

*  See  El.  Dif.  Eq.  \  5. 


50  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§  17,  18 

is  a  solution  of  (17)  ;  hence, 

(33) 


where  <£(.Y,  y)  =  const,  is  any  selected  form  of  solution  of  (17).     Re 
arranging  the  terms  in  (33), 

& 


where  p(x,  y)  is  the  common  value  of  the  two  fractions.     Whence 


(35)  • 

Conversely,  if  U^f  leaves  the  differential  equation  unaltered,  U.,f 
given  by  (35)  will  also  do  so,  no  matter  how  /p(^>)  and  p(x,  y)  may 
be  chosen  (it  being  understood  throughout  that  all  functions  involved 
are  to  be  generally  analytic).  For,  by  hypothesis,  using  (31) 


then         (U,A}f  =  (F($}U»  A)f+(pA,  A)f 

J+  P(AA)f- 


Hence  every  gn>i/f>  which  leaves  the  differential  equation  unaltered 
is  given  by  (35),  U\f  being  one  group  of  tins  sort. 

If  F($]  is  a  constant,  the  resulting  group  gives  the  same  integrat 
ing  factor  as  U\f. 

If  F(<f>)  is  identically  zero,  the  resulting  group  is  trivial  (§  12). 

18.  Differential  Equations  Invariant  under  a  Given  Group.  —  In 
order  to  make  use  of  the  theorem  of  §  12,  a  group  leaving  the  differ 
ential  equation  unaltered  must  be  known.  While  such  groups  always 


§  iS         DIKKKKKNTIAI.    F.<  H'ATIOXS    OK   TIIK    KIKST    ORDKK  51 

exist,  and  are  sometimes  suggested  by  the  nature  of  the  problem 
giving  rise  to  the  differential  equation,  the  number  of  equations  for 
which  they  are  known  is  comparatively  small.  The  converse  prob 
lem  of  finding  the  general  type  of  the  differential  equations  invariant 
under  a  given  group  is  much  more  direct.  And  while  its  complete 
solution  requires  the  knowledge  of  the  path-curves  of  the  group  and 
usually  one  or  several  quadratures,  it  is  practicable  to  supply  these 
in  a  large  number  of  cases  of  interest. 

It  is  clear  that  the  differential  equation  obtained  by  equating  an 
invariant  of  the  extended  group  (§  13)  to  an  arbitrary  constant  is 
invariant.  The  general  type  of  invariant  of  the  extended  group  is 
obtained  by  taking  an  arbitrary  function  of  two  independent  solu 
tions  of  ([9],  §  n) 


Passing    to    the    corresponding    system    of    ordinary    differential 
equations 

dx          d  dl 


t(x,y)     !,(*,?)      V( 
the  first  equation  is  recognized  as  (n),  §  7.     Its  solution  is 


A  second  solution,  independent  of  this  one,  must  involve  /.     Writ 

ing  this  in  the  form 

U'(x,y,/)**  COnst*, 

the  general  solution  of  (36)  will  be  of  the  form/(//,  //').  Equating 
this  to  an  arbitrary  constant  gives  the  general  type  of  invariant  dif 
ferential  equation.  There  is  ho  loss  of  generality  in  equating/(//,  //') 

*  Since  u'(x,yty')  is  an  invariant  of  the  extended  group  /  "/  and   involves;'',  it  is 
known  as  a  ///-.*•/  Jiffo  cntial  invariant  of  the  yroup  I  Y. 


52  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§18,19 

to  zero,  the  arbitrary  constant  being  incorporated  in  the  arbitrary 
function/.     So  that  the  general  type  bf  invariant  equation  is 

(38)  /(*,i 


Several  methods  for  rinding  a'  suggest  themselves  : 

(a)   Solving  u(x,  y}  =  c  forj,  and  replacing  it  by  its  value  in  terms  of  x  and 
c  wherever  it  occurs  in  the  first  and  last  members  of  (37),  the  Riccati  equation 


(39)  =         +_ 

d*     td*     t\dy     fe  By 

results.     In  Note  II  of  the  Appendix  it  is  shown  that  this  equation  can  be  solved 
by  quadratures. 

(Z»)  The  introduction  of  canonical  variables  (which  can  be  found  by  a  quad 
rature  when  u  is  known,  §  10)  reduces  the  invariant  differential  equation  to  the 
simple  form 


dy  __  d*     dy      _ 

~~     (  }' 


as  will  be  shown,  I,  §  19.     Since  the  one  canonical  variable  x  is  the  invariant  // 
or  a  function  of  it  (§  10),  F(x}  is  a  function  of  u.     Because  of  the  general  type 


of  invariant  differential  equation  (38),  -  -  2,  —  may  be  taken  as  u1. 

dx+fc  v' 
d*      dy*      % 

(c}   Frequently  some  special  method  (see  El.  Dif.  Eq.  §  65)  may  be  found 
which  is  more  direct. 

19.   Illustrations  and  Applications. 

I.    Uf=  .   .     £  =  o,  y=  i.     .-.  rj'  =  o.     Equations  (37)  are 

dx  _  dy  __  dy' 

o  ~~  i  ""  o 

.-.  //  =  x,  ?/'  =y.     Hence  the  general  type  of  differential  equation 
invariant  muter    Uj  r=  v-  tsf(x,  y')  =  o,  or  y*  =  F(x). 


§I9        DIFFERENTIAL   EQUATIONS  OF  THE   FIRST  ORDER          53 

This  equation  is  characterized  by  the  absence  of  y.  The  variables 
are  separated  when  the  equation  is  solved  for  /. 

I'.  6^=  -.  It  is  readily  seen*  that  the  general  type  of  differ 
ential  equation  invariant  under  this  group  (of  translations  in  the 
direction  of  the  axis  of  x)  is  y'  =  F(y). 

This  equation  is  characterized  by  the  absence  of  x.     The  variables 

are  separable,  thus  -— —  —  dx. 

II.     ^=_JK|?+*J£.        £=-X,    rj=X.        /.V=l4-/2. 

' 

dx       dy         dy' 
Equations  (37)  are 


-y       x       i+/2 

/.  u  =  jc+y2.  To  find  //,  multiply  numerator  and  denominator  of 
the  first  member  by  —  y,  and  those  of  the  second  member  by  x  ;  then 
by  composition  (Et.  Dif.  Eg.  §  65,  3°), 

x  dy  —y  dx        dy1 


r+72' 


.*.  «'  =  tan"1-    —  tan~Ty.     It  is  simpler  to  take  the  tangent  of  this 

"v  _    -  •' 

function  as  the  second  invariant;  i.e.  «'=-^-  --  -—.      Hence  the  gen- 

x+tf  d,     ^  d, 

eral  type  of  differentia!  equation  invariant  under  Uf=  —  y  -r  —  ^XJT 


*  It  is  suggested  as  an  exercise  that  the  student  actually  carry  out  the  work  here 
and  in  the  cases  below,  where  results  alone  are  given. 

Of  course,  the  differential  equation  invariant  under  a  group  whose  number  is 
primed  may  be  obtained  from  that  invariant  under  the  corresponding  unprimcd  one 
by  interchanging  x  and  r  and  v'  and  '  .  I'.ut  as  an  attempt  is  being  made  here  to 

make  a  collection  of  differential  equations  invariant  under  known  groups,  the  foims 
by  which  these  differential  equations  are  most  readily  recognized  are  given. 


54  THEORY   OF  DIFFERENTIAL   EQUATIONS  §  19 

Note. — This  form  of  the  invariant  differential  equation  is  obvious 
from  geometrical  considerations,  since  //  is  the  square  of  the  radius 
vector  to  any  point  on  an  integral  curve,  and  //'  is  the  tangent  of  the 
angle  between  the  radius  vector  and  the  tangent  to  the  curve.  Since 
any  function  of//  and  //',  containing  v',  can  be  used  as  a  first  differ 
ential  invariant,  \/—  — -  or  — ^  is  available.  So  also  is 

Vi  +/2 

These  are  respectively  the  distance  of  the 

normal  and  that  of  the  tangent  from  the  origin,  each  of  which  is  left 
unaltered  by  the  group  of  rotations  about  the  origin.  Hence  the  gen 
eral  type  of  differential  equation  invariant  under  this  group  may  also 
be  written 


III.  Uf=y-^-.     £  =  o,  77  =y.      .'.  T/  =y.      Equations  (37)  are 

<&     <ty_ 

o  ~~  y 

Hence   the  general  type  of  differential  equation 
invariant  under  irf~y^  W/f*i  ~j,  or  }~  =  F(x). 

This  equation  is  characterized  by  being  homogeneous  in  y  and  y\ 
The  variables  are  separated  when  the  equation  is  solved  for  -^ . 

*•.  f  J 

III'.  Uf=.  Xjf-*  It  is  readily  seen  that  the  general  type  of  differ 
ential  equation  invariant  under  this  group  (of  affine  transformations) 
is  xy'  =  F(y).  .  ^ 

The  variables  are  separable,  thus    _     -  =  — . 
1  /M  .r)       x 

IV.  Ufi~  .x -!  +)?{-.     Here  V  =  o,  and  »  =-r,  //'  =  v'.     Hence 

av        By  A- 


§I9         DIFFERENTIAL    EQUATIONS   OF   THE    FIRST   ORDER  55 

the  general  type  of  differential  equation  invariant  under 


This  equation  is  characterized  by  being  homogeneous  in  x  and  y. 

Note.  —  An  equation  M  dx  +  N  dy  —  o  is  of  this  type  when  M  and 
TV  are  homogeneous  functions  of  x  and  y,  and  of  the  same  degree. 
In  this  case  the  integrating  factor  of  §  12  is  —  —  —  •  (Compare 


•-.       Herei/=E  —  2 /,   and    u  =  xyt    //' =  A~y. 

Hence  the  general  type  of  differential  equation  invariant  under 

t\f        f) f  * 
Uf^  x  :    —  v  ;     is  f(x\\  .vV )  =o,  or  AT'  =  r/^  (AJ). 

This  equation  is  characterized  by  being  homogeneous  in  A*,  }',  y\ 
when  these  elements  are  given  the  weights  i,  —  i,  —  2  respectively. 
(Compare  VI  below.) 

Note.  —  An  equation  M  dx  -f-  N  dy  =  o  is  of  this  type  when  J/= 
(AT),  ^=A;/!,(AT).     In  this  case  the  integrating  factor  of  §  i  2  is 

(Compare  Rl.  Dif.  Eq.  §  17.) 


VI.      Uf^x       +  nv       -*     f  =  A-,  w=  ;/r.      .'.  «'=(//  -  Or',  and 
c'Av         "    fJr 

//  =  ••-,  w's-^—j-      Hence   //(/•  general  type  of  differential  equation 
x  x 

*  n  may  be  any  number.     In  particular  //  —  i  gives  IV,  while  >t  i  ^ivcs  V,  and 

n  —  o  gives  III'. 

If  the  group  be  written  in  the  more  symmetrical  form     I  r/~a.v  Q-  -\-  Iry    •      t}1R 

c).»-  (>' 

invariant  differential  «quation  takes  the  fonn.vr'      r  /•'(•''  )•    a-    /•  gives  IV,  a=  -  b 

\  \-h/ 

gives  \r,  ,/       o  gives  1  1  I,  />      o  gives  1  II  '. 


56  THEORY    OF    DIFFERENTIAL   EQUATIONS  §  19 

invariant  under    Uf=x-£--\-nv--   /V/(  — ,  ——  ]  =  o. 
da*       "  dv          vv"     A-"  V 


This  equation  is  characterized  by  being  homogeneous  in  x,  y,  y1 
when  these  elements  are  given  the  weights  i,  «,  n  —  i   respectively. 

Thus  the  differential  equation 

xy-y1'2  —  yy  +  x  =  o 

comes  under  this  head;  for  giving  x,  y,  y'  the  weights  I,  n,  n  —  I  respectively, 
the  separate  terms  have  the  weights  i  +  2  n  +  2  n  —  2  or  4  ;/  —  i,  3  n  +  n  —  i 
or  4  n  —  i,  i  respectively.  These  are  equal  to  i  if  n  =  %.  Hence  the  differen 
tial  equation  is  invariant  under  the  group 


d*         by 

VII.      Uf=  4>(x)  ¥•     £  =  o,  rj  =  4>(x).     .'.  T/  =  <f>'(x),  and  //  =  .v, 

//'  =  <j>(x)y'  —  <j>'(x)y.     Hence  the  general  type  of  differential  equation 
invariant  under 


Uf=  +(X)        is  f[x,  <}>(x)y'  -  #(X)y\  =  o,  or  y'  ~ 

This  equation  is  characterized  by  being  linear  in  y  and  y'. 

Note.  —  Using  the  usual  notation  for  the  linear  equation 

y'  +  P(x)y=Q(x\ 

the  group  which  leaves  it  unaltered  is  Uf=e~lrdt-^-     The  integrat 
ing  factor  of  §  12  is  e!r'u.     (Compare  I',  I.  /)//.  Eq.  §  13.) 

VII'.      Uf=\l>(y)^--     It  is  readily  seen  that  the  general  type  of 
ox 


§  i9        DIFFERENTIAL   EQUATIONS   OF  THE   FIRST   ORDER          57 

differential  equation  invariant  under  this  group  is 

i       i//(y)    \  dx      i//(v) 

v,     ,  —  *-)*•*  x\  =  Qt  or  -  x  =  F(y\ 

y  *w  y       •*   »oo 

This  equation  is  linear  in  x  and  —  • 

dy 

VIII.      f.y=\f/(}')  —  •      The  general  type  of  differential  equation 
invariant  under  this  group  (which  includes  III  as  a  special  case)  is 


In  this  equation  the  variables  are  separated. 

VIII'.      Uf=<l>(x)  -     •      The  general  type  of  differential  equation 
ox 

invariant  under  this  group  (which  includes  III'  as  a  special  case)  is 
y^(x)  =  F(y\ 

The  variables  are  separable. 


IX.      Uf=  4> 

dy 


.-.  V  =  4>'(x)t(J)  4-  ^ W^'O')/-     Equations  (37)  are 
dx_        dy  dy' 


o       f(*] 

/.  u  =  x.     u'  may  be  obtained  by  solving  the  linear  equation 


in  which  x  is  treated  as  a  constant.     An  integrating  factor  is  - 


58  THEORY   OF-'   DIFFHRKMTIAL   EQUATIONS  §19 

Hence  the  ge  neral  type  of  differential  equation  invariant  under 


The  transformation  v  =  (  -^—  reduces  this  to  the  linear  equation 

J 


dx 


Note 


.  —  In  particular,  if  ty(y)  is  y,  Uf=  <^>(jc)y    *  leaves  unaltered 


the    equation    /+;  —  -ry=fF(x).       Hence    the   Bernoulli 


equation  ^-+Py=  Qy  is  invariant  under  the  group  Cy=fel(s-})1'''£^-' 
dx  dy 

J(l-^)/></x 

The  integrating  factor  of  §  12  for  this  equation  is  -  •     (Com 

pare  EL  Dif.  Eg.  §  14.) 

IX'.      Uf=  *t>Mt(y)  —  •      The  general  t\pc  of  differential  cqua- 
dx 

..  .  ,      .,  .  i     dx      \lf'(y]  C  dx         7V  N 

twn  invariant  under  this  group  is  -  —  =  /"(  v). 


Considering  y  as  the  independent  variable  in  this  equation,  the 

latter  is  reduced  to  the  linear  form  by  the  transformation  /=  I    '  A   • 

J      > 

X.      U= 


u'  is  easily  found  by  method  (li)  of  §  18.     The  canonical  variables 

are  dx  dx      .rv'-//i' 

'' 


Hence  the  general  type  of  differentia!  equation  invariant  under 


§i9         DIH'KRENTIAL    KnUATlo.NS   OF   THE    FIRST   ORDER          59 

Note,  —  Several  particular  cases  are  of  special  interest  :  — 
i°  If  <b(x}  —  x\  the  general  type  of  differential  equation  invari 
ant  under 


y 

I  lence   xy1  —  ny  =  _r*/M  -^  -  j    is  invariant  under 


uf~  x*-kx   -  -f 

dx 


The  Riccati  equation 


dv 
x      — 


conies  under  this  head  when  n  =  2  a  ;  for  in  this  case 


(Compare  Boole,  Differential  Equations,  p.  92  ;  Forsyth,  Differential 
Equations,  §  109.) 

2°  If  <j>(x)  =  xrt  n=  i,  the  invariant  differential  equation  reduces 

to  .vr' —  v  =  .r1~r/M  ~   ]•     The  right-hand  member  is  simply  a  homo 
geneous  function  of  A-  and  y  of  degree  i  —  r.      Hence  a  differential 

equation  of  the  form     v  -—  .vr'  =  .vA'/''f  -  ],   where  the  right-hand  incni- 

\*/ 

her  is  a  Jiomogeneous  function  of  x  and  y  of  degree  k,  is  invariant 


dx         dy 
The  integrating  factor  of  §  12  is  -     — -•     (Compare  El.  Dif.  Eq. 


60  THEORY   OF   DIFFERENTIAL   EQUATIONS  §19 

3°  If  4>(x)  =  xr,  n  =  —  i,  the  invariant  differential  equation  reduces 
to  xy'  +  jr  =  x~l~rF(xy),  or  xy'  + y  =yl+rl?(xy).  Hence  a  differential 
equation  of  the  form  xy1  -\- y  =  ykF(x})  is  invariant  under  the  group 


The  integrating  factor  of  §  1 2  is  —      - — - ,  a  well-known  fact. 


X'.    £7  =  i/r 0') f  x  ~-  +  ;y;  y;  J  •      77/<f  ^«<?r«/  /v/te  <?/  differentia 
equation  invariant  under  this  eroup  is  xv'  —  nv  =  —r-rF[  —  ]• 

•AC;1)    \**y 

i°  If  \l/(y)  =}'*,  this  differential  equation  reduces  to 


2°  If  i//0')  =}'*,  n=  i,  the  differential  equation  takes  the  form 

rN 


Hence  a  differential  equation  of  the  form 

y  —  xy'  =y'  [a  homogeneous  function  of  x  and  y  of  degree  k~\ 

is  invariant  under  the  group    Uf=\*~k(  x  -  —  Vyjry 


3°  If  ;;=—i,   ^(y}=y*,    the    differential    equation    reduces    t 
xy'  +y  =  y'xg+lF(xy).     Hence  a  differential  equation  of  the  form 


dv 


§i9         DIFFERENTIAL   EQUATIONS   OK  THE   FIRST   ORDER          6l 

The  student  should  show  that  the  following  groups  leave  the  corre 
sponding  differential  equations  unaltered  : 


XI. 

\ 


XII.     Uf=  a    ~  +  b-,  y  =  F(bx -  ay). 

ox        oy 


using  method  (r),  §  1 8, 

,,„' 

'-A 


x  —  vv' 


using  method  (£),  §  1 8. 

fdi 
XIV.       67=<£( 

XIV.       Uf=ti( 


xv.    iy^     ±r      r. 

dx      7   dy 


xvi.    uf=  ^ 

«,WO*y  4-  A+  v')  = 
where  M  =  ^'^  v=  C^(x)itx. 

*  This  group  is  characterised  l>y  liaving  ^  a  function  of  \  only,  and  77  a  linear  func 
tion  of  y.  It  is  mentioned  by  1'rotes.sor  Dickson,  />'////<•////  of"  the  An:.  Math,  .Si'<  ., 
Vol.  V.  p.  453. 


62  TIIKORY   OF    DIFFFKFXTIAL    EQUATIONS  §19 

XVI'.     Uf=  <M 


where 


S***1  'Xv,  o-  EEj 


Remark.  —  When  a  differential  equation  is  recognized  as  coming 
under  several  of  the  above  heads,  and  the  corresponding  integrating 
fastors  are  distinct,  the  solution  of  the  differential  equation  is  obtained 
at  once  by  equating  the  quotient  of  two  distinct  integrating  factors 
to  an  arbitrary  constant  (§  16). 

Thus  the  differential  equation 


is  linear.     Hence,  from  VII,  the  group  Uf=x-J~  leaves  it  unaltered, 

i          dy 
and  gives  the  obvious  integrating  factor  —  • 

But  it  is  also  readily  seen  that  each  term  of  the  equation  is  of  the 
weight  ;-  when  x,  y,  y'  have  the  weights  i,  r,  r  —  i  respectively  ;  hence, 

from  VI,  the  group  Uf=x—-+ry  {-  leaves  the  equation  unaltered, 

dx          dy 

and  gives  the  second  integrating  factor    -  —  •     The  solti- 

(r—  i>v>'  —  x 
tion  of  the  equation  is  therefore" 

/          Ny         ,.  , 
=(r  —  i  V-  —  x      =  const. 

x 


It  may  be  noted  that  the  equation  also  comes  under  X,  2°,  and  is 
erefore  invariant  under   Uf==^ 

iL 

previously  found  integrating  factor  - 


therefore  invariant  under   Uf=x-  '  •    +xl~ry  —  •     This  leads  to  the 

(Av  dy 


§§i9)2o     DIFFERENTIAL   EQUATIONS  OF  THE  E1RST  ORDER       63 

'  * 

As  another  illustration  of  a  class  of  equations  obviously  invariant 
under  several  distinct  groups,  the  equation 

•*//  —  /+1  =  xr  or  xy'  -  y  =    - 


may  be  mentioned.     Under  the  head  of  VI  it  is  readily  seen  to  be 

invariant  under  Uf=  (r-f  \}x-+--\-  ry-—  :  as  a  Bernoulli  equation,  IX, 

dx          dy 

Tr+l   fif 

it  is  invariant  under  Uf=*        —  •     From  these  its  solution  is  found 

y  ty 

at  once  to  be 

»  ?  = 


This  equation  also  comes  under  X,  2°. 

20.    Second  General  Method  for  Solving  a  Differential  Equation. 
Separation  of  Variables.*  —  The  simple  form  of  the  differential  equa 

tions  invariant  under  the  group  of  translations  6^=—  (I,  §  19)  sug 

gests  as  a  practical  method  for  solving  a  differential  equation  invari 
ant  under  a  known  group  the  introduction  of  canonical  variables 
(§  10).  The  reduction  of  the  group  to  the  canonical  form  reduces 
the  differential  equation  to  the  form 


in  which  the  variables  are  separated.     The  solution  is  then  obtained 
by  the  quadrature  ,-• 

y  =  J  F(X)JX  +  c. 

Finally  it  is  necessary  to  pass  back  from  the  canonical  variables  to 
the  original  ones. 

*  This  method  was  discovered  by  Lie  in  1869,  thus  antedating  the  method  of  J  12  by 
live  years.  1  listorically  it  is  ot  interest  because,  it  is  the  first  known  method  of  integra 
tion  which  makes  use  of  the  invariance  ot"  a  differential  equation  under  a  group. 


64  THEORY   OF   DIFFERENTIAL   EQUATIONS  §  20 

Since  the  differential  equation  invariant  under  Uf=.  —  *(  I',  §  19) 

WP 
is  of  the  form  — ^—  =  dx,  the  reduction  of  the  group,  under  which  a 

differential  equation  is  invariant,  to  this  form   also  enables  one  to 
separate  the  variables  in  the  differential  equation. 

While  either  of  the  above  transformations  brings  the  differential 
equation  into  a  very  simple  form,  the  actual  introduction  of  canonical 
variables  into  the  differential  equation  and  the  final  passing  back  to 
the  original  variables  may  not  prove  as  simple  as  in  the  case  of  other 
variables  that  could  be  used  to  equal  advantage.  Thus,  for  example, 

if,  in  the  group  £7=  £^  +  17-^-  which  leaves  the  differential  equation 
uoc       ^y 

unaltered,  £  is  a  function  of  x  only,  the  introduction  of  the  new  vari 
ables  (§  9) 

x  =  x,   y  =  u(x,y) 

reduces  the  group  to  the  form 


whence  the  differential  equation  must  take  the  form  (VIII',  §  19) 

(40)  t(x)y'  =  F(y\ 

in  which  the  variables  are  separable  at  once. 

This  set  of  variables  works  especially  well  in  the  case  of  two  perfectly  well- 
known  classes  of  differential  equations,  and  leads  to  the  usual  methods  for  solv 
ing  them : 

i°  The  homogeneous  equation 

Mdx  +  Ndy  =  o, 

*  Owing  to  the  complete  symmetry  of  the  two  groups   fTf---    •     and   Uf~-    ,  we- 

dx  fly 

shall  sny  that  the  group  in  either  case  is  in  the  canonical  form,  and  the  variables  that 
reduce  a  group  to  either  form  will  be  said  to  be  the  canonical  variables  of  the  group. 


§20        DIFFERENTIAL   EQUATIONS   OF  THE   FIRST   ORDER         65 

where  M  and  TV'  are  homogeneous  and  of  the  same  degree,  is  left  unaltered  by 

the  group  (IV,  Note,  §  19) 

Uf=lXg.+yV. 

d*        dy 

V 
The  new  variables  x  =  x,  y  —-  —  reduce  the  group  to  the  form 


whence  the  differential  equation  assumes  the  form   (40),  an-d  the  variables  are 
separable.     (Compare  EL  Dif.  Eq.  §  10.) 

2°  The  equation 

-  o 


is  left  unaltered  by  the  group  (V,  Note,  §  19) 


Hence,  the  new  variables  x  —  x,  y  —  xy  reduce  the  equation  to  the  form  (40) 
in  which  the  variables  are  separable.     (Compare  EL  Dif.  Eq.  §  12.) 

In  an  analogous  manner,  if  rj  is  a  function  of  y  only,  the  introduc 
tion  of  the  new  variables 

x  =  u(x,y),   y=y 

reduces  the  group  to  the  form 


whence  the  differential  equation  must  take  the  form  (VIII,  §  19) 


in  which  the  variables  are  separated. 

More  generally,  if  <f>(x)  and  \l/(y),  any  functions  of  the  respective 
canonical  variables,  are  taken  as  new  variables,  it  is  readily  seen  that 
the  resulting  differential  equation  will  have  its  variables  separated. 
In  certain  cases  such  forms  can  be  chosen  for  these  functions  as  to 
simplify  the  actual  work  required  in  introducing  new  variables. 


66  THEORY   OF  DIFFERENTIAL   EQUATIONS  §§  20,  21 

Remark.  —  It  is  interesting  to  note  that  the  knowledge  of  a  group, 
under  which  a  given  differential  equation  of  the  first  order  is  invari 
ant,  enables  one  to  find  both  an  integrating  factor  (§  12)  and  a  set 
of  variables  which  are  separable  in  the  transformed  equation.  (Com 
pare  El.  Dif.  Eq.  §  17.) 

The  integrating  factor  can  be  written  down  at  once  when  the  dif 

ferential  equation  has  been  solved  for  ^-,  or  what  is  the  same  thing, 

dx 

when  it  has  the  form  M  dx  -\-  N  dy  =  o. 

To  find  the  new  variables  that  are  to  be  separable,  the  solution  of 
another  (frequently  simple)  differential  equation  of  the  first  order 
(giving  the  path-curves  of  the  group)  and  usually  one  or  several  quad 
ratures  are  necessary. 

In  actual  practice,  neither  method  should  be  insisted  upon  to  the 
exclusion  of  the  other.  In  Table  I  of  the  Appendix  will  be  found  a 
list  of  the  more  commonly  occurring  and  easily  recognizable  classes 
of  equations  of  the  first  order,  and  methods  for  solving  them. 

21.    Singular  Solution.*  —  Let 

(25)  /(*,.?,  y>=o 

be  an  invariant  differential  equation  under  the  non-trivial  group 


Its*  family  of  integral  curves  being  left  unaltered,  as  a  whole,  if  this 
family  has  an  envelope,  the  latter  must  be  an  invariant  curve  of  the 
group  ;  moreover,  it  is  a  path-curve,  since  the  group  is  supposed  to  be 
non-trivial,  thus  interchanging  the  integral  curves  among  themselves. 
The  equation  of  the  envelope  being  a  singular  solution  of  the  dif 
ferential  equation  (EL  Dif.  Eq.  §  30)  the  value  of  its  slope/  -at  each 

*  Tli  is  section  is  based  on  an  article  by    |.  M.  Page,  entitled  "  Note  on   Singular 
Solutions"  in  tin;  American  Journal  of  Mathematics,  Vol.  XVI  II,  p.  95. 


§21         DIM  KKKXTIAL    K.nrATlONS   OK   THE    KIKST   ORDER  67 

point  (xfy)  must  satisfy  (25).     Since  the  slope  of  a  path-curve  at  the 

point  (x,  y)  is  ^  •  *-  /,  the  equation  of  the  envelope  must  be  con 
tained  in  *          "f?  ;>' 

(41) 

Remark.  —  In  the  above  process  (41)  was  found  as  the  equation 
of  a  path-curve  which  satisfies  the  differential  equation.  If  a  par 
ticular  integral  curve  happens  to  be  a  path-curve  of  the  group,  its 
equation  is  also  included  in  (41).  But  all  extraneous  loci,  ^uch  as 
nodal,  cuspidal,  and  tac-loci  (/?/.  Dif.  Eq.y  §  33)  which  may  be  path- 
curves  but  are  not  solutions  of  (25)  will  not  be  included  in  (41). 

Ex.  1.   xy-y's  —  jrV  +  x  =  o. 

This  equation  is  invariant  under  Uf=2x-~  —  \~)'"^  ,  0%  §  £9)- 
Its  general  solution  is  s-x2  —  cf  +  i  =  o. 

Replacing  v',  wherever  it  occurs  in  the  differential    equation,  by 

-   gives  lT(4  x*  —  y4)  =  o. 

x  =  o  is  a  particular  solution  for  c  =  oo  . 
4  x2  —  j'4  =  o  is  the  singular  solution. 

Ex.    2.    (i+A:V2=i- 
This  equation  is  invariant  under  Uf=:~.     (I,  §  19.) 

y  =  -  =  oo.     In  this  case,  writing  the  differential  equation  in  the  form 


—  =  o  gives  the  singular  solution  i  -f-lr2  =  o. 

Kx.   3.    .xy-'-.vr'-v-o.  (VI,  §  19.      «  =  -2.) 

'I-:x.   4.    r/;rr''-'  -    2  .vr'  +.v  =  o.  (IV,  §   19.) 


68  THEORY    OF    DIFFERENCIAL    EQUATIONS  §21 

Ex.  5.   /3  —  4  ATI-'  +  Sy2  =  o.  (VI,  §  19.     n  =  3.) 

Ex.  6.    v  =  2  AT'  +/y3.  (VI,  §  19.     w  =  «.) 

Ex.   7.   xy-  +  ^''  +1  =  0.  (VI,  §  19.     n  =  ~  J.) 

It  is  suggested  as  an  interesting  exercise  that  the  student  examine, 
in  the  light  of  the  Lie  theory  as  presented  in  this  chapter,  the  vari 
ous  examples  involving  differential  equations  of  the  first  order  to  be 
found,  for  example,  in  Chapters  II,  IV,  V  of  the  author's  Elementary 
Treatise  on  Differential  Equations. 


CHAPTER   III 


MISCELLANEOUS   THEOREMS   AND   GEOMETRICAL   APPLI 
CATIONS 

22.    New  Form  for  Integrating  Factor.  —  In  §  12  it  was  seen  that 

i 

V  -n 

R 


is  an  integrating  factor  for 
Mdx  +  Ndy  =  o 

if    the    latter   is    invariant 
under 


FIG.  i 


Lie,  by  purely  geometrical  considerations,  gave  a  new  form  *  to  this 
factor,  which  is  not  only  interesting  but  also  useful  in  certain  classes 
of  problems.  In  Fig.  i,  let 


be  some  one  of  the  integral  curves  of  the  differential  equation.  The 
infinitesimal  transformation  of  the  group  transforms  this  into  an 
infinitely  near  curve  of  the  family 


by  transforming  any  point  (x,  y)  of  it  into  (.v  -f-  £  &a,  y  -\- 

/'/',  —  the  distance  between  these  points  is  V£"  +  17"  &*. 


First  published  in  the  Gesellschaft  der  Wissenschaften  zu  Christiania,  1874. 

69 


70  THEORY   OF   DIFFERENTIAL   EQUATIONS  §22 

M 

The  slope  of  the  tangent  at  P  is  --  .    If  T  is  the  point  (x  —  Nt 

y  +  M),  the  length  of/'7'is  -\/M'-+~N*t  and  the  area  of  the  paral 

lelogram  PTRPi  is  (£M+rjlV)8a,  or  —  . 

P 
Let  8;z  =  PN,  the  length  of  the  normal  to  the  first  curve  at  P, 

intercepted  by  the  second  curve  ;  this  is,  to  within  infinitesimals  of 
higher  order  than  the  first,  equal  to  PQ,  the  altitude  of  the  above 
parallelogram.  Hence 


or 

(2}  =•- 

= 


This  form  of  the  integrating  factor  is  serviceable  in  the  case  of  an 
interesting  class  of  differential  equations  : 

If  the  integral  curves  of  a  differential  equation  are  known  to  be 

^ 

a  family  of  parallel  curves*  for  which  —  is  constant  all  along  each 

6a 

one  of  the  curves,  it  follows  at  once  from  (42)  that 

U»0 

is  an  integrating  factor.  The  involutes  of  a  curve,  which  are  the 
orthogonal  trajectories  of  the  tangents  to  the  curve,  are  known  to 
form  a  family  of  parallel  curves.  Hence  an  integrating  factor  of  the 
form  (42')  is  known  at  once  for  their  differential  equation. 

Ex.    Find  the  involutes  of  the  circltj  x-  -f-  y~  =  I. 

The  differential  equation  of  the  tangents  to  the  curve  is  f  writing/  for 


*  Two  curves  are  said  to  be  parallel,  if  the  distance  between  them  measured  along 
the  normal  to  one  of  them  is  constant  all  along  the  curve.  (In  this  case,  it  is  well  known 
that  the  normal  to  either  curve  is  normal  to  the  other.) 


§22  MISCELLANEOUS  THEOREMS 

Hence  the  differential  equation  of  the  family  of  involutes  is 


71 


(xy  -f  -V 
The  integrating  factor  given  by  (42)  is 


—  l   d   -  o. 


To  integrate  the  exact  equation 

(.vr  -f  V**  -f  ^  - 


one  may  proceed  in  the  usual  way  (see  El.  Dif.  Eq.  §  8)  to  integrate 


where  ^'  is  considered  a  constant.      Multiplying  numerator  and  denominator  by 
x  —  y\fx*  -f-  yfi  —  I,  this  becomes 


r  .'••"•    - 

J     ,\--    r  _r'J 
Letting  .r2  -f  r-  =  /, 


=       i    gin_1   _ 


=r  x/.r-  +  r-  —  I  +  sin'1  -        —  . 


Hence,  the  equation  of  the  family  of  involutes  is 


—  I  -f  sin~!  -  —  tan"1  *  =  const. 

^  .,-'  +  ^2  JT 


72  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§  23,  24 

Remark.  —  From  the  nature  of  the  problem,  it  is  evident  that  the  family  of 
involutes  is  invariant  under  the  group  of  rotations  &/=  —  y  -.  -  +  x  -*£-•       Hence, 

the  methods  of  §§  12  and  20  are  also  applicable.  It  is  readily  seen  that  the  inte 
grating  factor  given  by  the  method  of  §  12  is  the  same  as  that  found  in  the  text. 
The  method  of  §  20  should  be  carried  out  as  an  exercise. 

23.    Two  Differential  Equations  with  Common  Integrating  Factor. 
If  //,  is  an  integrating  factor  for  two  distinct  differential  equations, 


=  o>  and  M2  dx  +  N2  dy  —  o, 


_          =  o  and 


dy  dx  dy  dx 

^  1°S>  _  M  d]o£f'  =  **M± 

dx  dv  dy 


(A<>\  .• 

a  lo 


. 
I  O.T  dy  dy          djf 

Here  N^M^  —  N.M^^Q,  since  the  differential  equations  are  sup 

posed  to  be  distinct.     Hence  (43)  can   be   solved   for  -  —  sL£  and 

dx 

—  2!L£.  •  log/x  can  then  be  determined  by  a  quadrature,  and  /x  may 
dy 

be  obtained  at  once  from  this.      Hence  the 

THEOREM.  —  If  hvo  differential  equations  of  the   first  order   are 

known  to  have  a  common  integrating  factor,  the  latter  can  be  found 

* 
by  means  of  a  quadrature. 

24.  Isothermal  Curves.  —  A  family  of  curves  which,  together  with 
the  family  of  orthogonal  trajectories,  divides  the  plane  into  infini 
tesimal  squares,  is  called  a  family  of  isothermal  curves.  In  general, 

*  This  is  also  obvious  from  tin-  form  of  the  differential  equation,  when  cleared  of 
fractions,  viz.  :  x  +  yp  —  Vi  -f  /-.  (Sec  II,  Note,  $  19.) 


MISCELLANEOUS  THEOREMS 


73 


FIG.  2 


a  family  of  curves  and  their 

orthogonal  trajectories  divide 

the   plane   into   infinitesimal 

rectangles.       For,    selecting 

any     pair     of     neighboring 

curves,  /and  //(Fig.  2),  of 

the   one   family  it   is  always 

possible   to  find   a    pair,   A 

and  B,  of  the  second  family 

to     form      an     infinitesimal 

square  *  with  them  ;  besides, 

selecting  any  third  curve  /// 

of  the  first  family,  a  fourth 

curve  IV  can  be  found  such 

that  A,  /?,  ///,  IV  form  a  square  also ;  again,  selecting  any  third 

curve  C  of  the  second  family,  a  fourth  curve  D  can  be  found  such 

that  C,  Z>,  /,  //  form  a 
square.  But  with  these  se 
lections  made,  the  curves 
C,  D,  III,  IV  do  not,  in 
general,  form  a  square. 

Concentric  circles  are  read 
ily  seen  to  be  isothermal  curves. 
Their  orthogonal  trajectories 
are  the  straight  lines  through 
the  common  center  (Fig.  3). 
Any  pair  of  circles  of  radii 
r  and  r  +  \r  respectively 
(r>o)  form  an  infinitesimal 
FIG.  3  square  with  any  two  of  the 

*  This  curvilinear  quadrilateral  is  a  square  when  infinitesimals  of  higher  order  than 
the  first  arc  neglected,  the  length  of  arc  of  one  of  the  sides  being  taken  as  an  infinitesi 
mal  of  the  first  order. 


74 


THEORY   OF   DIFFERENTIAL   EQUATIONS 


§24 


straight  lines  which  intercept  the  length  Ar  on  the  inner  circle.  Moreover  these 
same  two  lines  form  squares^with  any  other  pair  of  circles  of  radii  kr  and 
^(r-fAr),  respectively,  k  being  any  constant  different  from  zero. 

From  the  definition  of  isothermal  curves,  8«(of  §  22)  can  be  made 
the  same,  at  any  point,  for  this  family  of  curves  and  for  that  of  their 
orthogonal  trajectories.     Moreover,  if  the  differential  equation  of  the 
one  family  is 
(17) 

that  of  the  other  is 
(17') 

Hence  the  two  equations  have  a  common  integrating  factor,  as  is 
evident  from  the  form  (42).  To  determine  this  integrating  factor, 
the  method  of  §  23  applies.  The  equations  (43)  take  the  form 


_  --  —  —     -  —  — 

dy          dy        ax 


log  /x     Nd  log  ^ 


dx 


dy 


dN 
~dj" 


dM 
dx 


whence 


(44) 


dJogjA 
dx 


dM 
-^. 
dy 


-= 
o 


.,  ,T 

M—  --  N-~ 


dx 


^ 
ox 


M*  +  N* 


dx 


dx 


Or 


dy 


dy 


dx 


Equations  (44)  are  interesting,  not  only  because  they  enable  one 
to  find  /A  by  a  quadrature,  but  also  because  they  lead  to  the  condition 


§24  MISCELLANEOUS  THEOREMS  75 

that  the  integral  curves  of  the  differential  equation  (17)  be  isothermal. 
For,  differentiating  the  first  of  (44)  with  r«pect  to_y  and  the  second 
with  respect  to  x,  and  equating 

(45) 

The  general  solution  of  this  is  * 

(46)  =  tan  [$(#  -f  iy)  -f  ty(x  —  iy)~], 

where  <£  and  ^  are  arbitrary  functions. 

The  condition  (45)  is  not  only  necessary  that  (17)  be  the  differen 
tial  equation  of  a  family  of  isothermal  curves,  but  it  is  also  sufficient. 
For,  when  M  and  ^  satisfy  (45),  equations  (44)  are  consistent,  hence 
a  common  integrating  factor  for  (17)  and  (17')  can  be  found.  But 
the  sum  of  the  squares  of  the  coefficients  of  dx  and  dy  is  the  same 
for  these  two  differential  equations.  Hence,  remembering  the  form 
(42),  S//  must  be  the  same  (to  within  a  constant  factor,  which  may 
be  made  unity  by  a  proper  choice  of  neighboring  curves)  in  the  two 
cases  at  any  point.  Hence  the  integral  curves  of  (17)  are  iso 
thermal  curves. 

Remark.  —  The  condition  for  isothermal  curves  in  terms  of  their 
finite  equation  and  that  of  their  orthogonal  trajectories  is  obtained 
in  Note  III  of  the  Appendix. 

i°  In  the  case  of  the  family  of  concentric  circles,  x1  +  y*-  —  const.,  the  differen 
tial  equation  is  x  dx  -\-  y  dy  =  o.  Hence  (45)  is  satisfied,  since  y2  tan"1-^  =  o. 

While  the  solution  of  this  differential  equation, ,as  well  as  that  of  the  differen 
tial  equation  of  the  orthogonal  trajectories,  y  dx  —  x  Jy  =  o,  is  very  simple,  it  is 
interesting  to  note  that  (44)  give  very  readily 

/I  2  x  dx  -f-  2  v  dy  ,,       /  o   .     ox 

</log/*  =  --  y    /  ^  =  -</log(>2+y*).     .-.  n  = 


This  is  the  common  integrating  factor  for  the  two  equations. 
*  See  El.  Dif.  Eq.  \  go. 


76  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§24,25 

2°   The  family  of  circles  tangent  to  the  axis  of  y  at  the  origin  x2  +  y-  —  ex  =  o 
has  for  differential  equation  ^c'2  —  y'2)dx  -f  zxydy  =  o.     It  is  readily  seen  that 

2  JCV 

V2  tan"1  -  *•  —  =  o;   hence  these  circles  form  an  isothermal  system.     The  dif- 
x--f 

ferential    equation    of  the    orthogonal    trajectories  is  2  xy  dx—  (x2  —  y~}<ty  —  o. 
While  this  is  easy  to  integrate,  it  is  worth  noting  that  (44)  give  /*  —  — 

(xi  +y2)2 

Moreover,  since  the  differential  equation  is  "  homogeneous,"  it  is  invariant  under 

the  group  Uf=x^-+y^L  (IV,  §  19).     Hence,  a  second  integrating  factor  is 
dx         dy 

(§  12)  /io  =  —        -  .     The  solution  of  the  equation  is  therefore  (§  16) 

=  const,  or  .r2  -f  y-  —  cy  =  o, 
/"2  y 

the  equation  of  the  family  of  circles  tangent  to  the  axis  of  x  at  the  origin. 


—  2 


Show  that  the  following  curves  are  isothermal,  and  find  their 
orthogonal  trajectories  : 

Ex.  1.    The  equilateral  hyperbolas  xy  —  const. 

Ex.  2.  The  similar  conies  ax1  4-  by*  =  const.,  when  and  only  when 
b  =  ±  a. 

Ex.  3.   The  coaxial  circles  through  the  points  (i,  o)  and  (—  i,  o), 


y 

25.  Further  Application  of  the  Theorem  of  §23.  —  An  obvious 
corollary  of  the  theorem  of  §  23  enables  one  to  find  an  integrating 
factor,  by  means  of  a  quadrature,  for  an  interesting  set  of  differential 
equations.  This  corollary  is  :  If  the  ratio  of  the  integrating  factors 
of  two  differential  equations  is  a  known  function,  the  integrating 
factors  can  be  found  by  a  single  quadrature.  For,  suppose  that 

(47)  &=4>(X,y) 

to 

is  a  known  function,  where  ^  and  /x2  are  the  integrating  factors  of 
J/!  dx  -f  TVi  dy  •=•  o  and  M.2  dx  4-  N*  dy  =  o 


§25  MISCELLANEOUS  THEOREMS  77 

respectively.     If  the  second  equation  be  written  in  the  form 

• 
<£  M.,  dx  +  $N»  dy  —  o, 

its  integrating  factor  is  also  fi\-  Hence  it  and  the  first  equation  have 
a  common  integrating  factor  and,  by  the  theorem  of  §  23,  this  can 
be  found  by  a  quadrature. 

Suppose,  now,  that  it  is  known  that  the  solutions  of  three  differen 
tial  equations  of  the  first  order 

J/!  dx  -\-JlVl{fy  =  o,    M*  dx  +  N»  dy  =  o,    M3  dx  +  Nz  dy  =  o 

can   be    made    to    assume    such    forms,    fa  =  const.,     fa  =  const., 

fa  =  const.,  that 

(48)  fa  =  fa  +  fa. 

If  Hi,  ft*,  /*3  are  their  respective  integrating  factors, 
dfa  =  n}(Mi  dx  +  NI  dy),    dfa  =  ^(M^  dx  +  N.2  <fy), 


Because  of  the  identity  (48) 

dfa  =  dfa  4-  dfa, 


or 
(49) 


whence  ^  =  M^i  +  ^  =  MI^  +  ^  and 


,  By  the  corollary  above,  H\  can  be  found  by  a  quadrature  ;  and  /u;, 
is  then  known  from  ('47').  After  finding  <£,  and  fa  by  a  single  quad 
rature  each,  fa  is  given  immediately  by  (48).  Hence  the 


78  THEORY   OF   DIFFERENTIAL   EQUATIONS  §25 

THEOREM.  —  If  it  is  known  that  the  solutions  of  three  differential 
equations  of  tJie  first  order  can  be  put  in  sitcJi  forms  <£[  —  const., 
<b.,  =  const. ,  d>.j  =  const.  tJiat  , 

= 


these  solutions  can  be  found  by  means  of  three  quadratures. 

This  theorem  has  some  interesting  applications  in  the  theory  of 
surfaces  *  : 

A.  If  the  rectangular  coordinates  of  any  point  (xty,  s,)  on  a  sur 
face  are  expressed  in  terms  of  the  parameters  u  and  ?',  the  expression 
for  the  element  of  length  of  arc  is,  using  the  usual  Gauss  notation, 

ds>  =  E  dir  +  2  Fdu  dv  +  G  dv\ 
where 

W£Y+(£Y+(£Y,  F=^*  • 8v8)'  • 8z  dz 

\vuj  du  dv 


dit  du         \du     '  d/  dv      du  d 


The  differential  equation  of  the  lines  of  zero  length,  usually  called 
minimal  lines,  is  then 

(50)  E  du*  +  2  Fdu  dv  -f  G  dir  —  o. 

This  differential  equation,  being  of  the  second  degree,  is  equivalent 
to  the  two  _ 


E  du  +    F+  ^F'2-EG  dv  =  o 
(51) 


E  du  +  (F-  V/^2  -  EG)  dv  =  o, 

which  are  essentially  distinct,  since  it  is  always  presupposed  that 
EG  —  //2  is  different  from  zero.  Let  «(//,  v)  =  const,  and  (3(u,  v) 
=  const,  be  the  solutions  of  (51).  These  are  the  equations  of  the 
minimal  lines.  Choosing  them  for  parametric  curves,  equation  (50) 

*  These  applications  will  be  of  interest  to  those  only  who  have,  at  least,  a  slight 
nrqiiiiintance  with  the  elements  of  Differential  Geometry.  They  have  been  taken  from 
Lie's  I'orlesungen  uber  Differentialgleichungen,  Chap.  9. 


§25  MISCELLANEOUS  THEOREMS  79 

takes  the  form  da  tip  =  o,  i.e.  E(a,  /3)  =  G(a,  ft)  —  o,  and  the  expres 
sion  for  the  element  of  length  of  arc  is 

</s-=2F(u,  P)  tin  tip. 
Introducing  the  new  parameters  i^  and  ?\  defined  by 


where    </>   and    «//   are    any   desired    functions    of    their    respective 
arguments,  tf  =  *(„„,.,)(,/„?  +  W), 

since  tiatiB  =  (ih<\  +  <h'\~)-     This    form    of   the    expres- 

V(*W(py  * 

sion  for  the  element  of  length  is  characteristic  of  isothermal  para 
metric  curves.     (Compare  Note  III  of  the  Appendix).     Hence, 

2  //!  =  U=  </>(«)  4-  \I>(P)  =  const. 
and  2  ivi  =  V=  <£(«)  —  \j/(fi)  =  const. 


are  the  equations  of  the  isothermal  curves  and  their  orthogonal  tra 
jectories,  respectively.  Since  <£(«)  =  const,  and  ty(J3)  =  const,  are 
equally  well  the  equations  of  the  minimal  lines,  it  is  evident  that  the 
identity  (48)  is  satisfied  by  the  equation  of  any  isothermal  system 
and  those  of  the  minimal  lines.  It  follows  then  from  the  theorem 
above  that  the  differential  equation  of  a  family  of  isothermal  cinves 
on  any  known  f  surface  can  be  integrated  l>\  means  of  quadratures. 
Besides,  the  knowledge  of  a  family  of  isothermal  lines  on  a  known 

*  In  the  case  of  a  real  surface,  a  and  /3  may  be  selected  as  conjugate  complex 
functions  of  u  ;,nd  v,  when  the  original  parametric  curves  are  real.  Real  isothermal 
curves  are  then  obtained  by  choosing  0  and  \f/  conjugate  functions  of  it.  and  /j 
respectively. 

t  A  surface  is  said  to  be  known  if  the  values  of  .r,^,  z  in  terms  of  the  parameters 
«,  -v  are  known,  or  if  the  forms  of  E,  F,  (7,  and  of  />,  D'  ,  D"  (to  be  introduced  below) 
are  given  in  terms  of//,  v.  In  this  particular  case  /:,  F,  G  only  need  be  known,  mini 
mal  anil  isothermal  lines  not  depending  upon  D,  /•>',  D"  . 


8o 


THEORY   OF   DIFFERENTIAL   EQUATIONS 


§25 


surface  enables  one  to  integrate  the  differential  equations  of  the  mini 
mal  lines  (51)  by  means  of  tu>o  quadratures. 

Remark  i.  —  For  surfaces  of  the  second  order,  surfaces  of  revolu 
tion,  and  minimal  surfaces,  the  lines  of  curvature  (see  B  below)  are 
known  to  be  isothermal  lines.  Hence,  in  the  case  of  these  surfaces 
the  differential  equation  of  the  lines  of  curvature  can  be  integrated 
by  means  of  quadratures. 

Remark  2.  —  In  the  case  of  a  minimal  surface  the  asymptotic 
lines  are  also  isothermals.  Hence,  on  such  a  surface  the  differen 
tial  equation  of  these  lines  can  also  be  integrated  by  means  of 
quadratures. 

B.  The  tangent  plane  to  a  surface  at  a  given  point  cuts  the  sur 
face  in  a  curve  which  has  a  double  point  at  that  point.  In  general, 
the  directions  of  the  tangents  to  the  two  branches  of  the  curve  at 
that  point  are  distinct.  In  this  way  two  directions  (in  general)  are 
determined  at  every  point  on  the  surface.  A  curve  on  the  surface 
whose  direction  at  every  point  coincides  with  one  of  these  directions 
is  called  an  asymptotic  line.  So  that,  in  general,  through  each  point 
on  the  surface  there  pass  two  asymptotic  lines.  The  differential 
equation  of  the  asymptotic  lines  is 


(5*) 
when 

j 
> 

d2x     dx     dx 
du2     du     dv 
d'2y      dy      dy 
dir     du     dv 
d2z      dz      dz 
du-     du     dv 

D  dir  + 
,    />'  = 

2  D'du  dv  +  D"( 
d*x       dx     dx 

fir  =  o, 

d2x  dx  dx 

du  dv     du     d?> 
d*y       dy      dv 
du  dv     du     dv 
d2z       dz      dz 

dv2  du  dv 
d2y  dy  dy 
dv-  du  dv 
d*z  dz  dz 

du  <>v     du     dv 

dv~  du  dv 

In  case  £>D"  —  D'2  =  o,  the  two  curves  coincide.  This  happens 
at  every  point  of  a  surface  where  the  Gauss  measure  of  curvature 
is  zero. 


§25  MISCELLANEOUS  THEOREMS  8 1 

Another  system  of  curves  playing  an  important  role  in  the  theory 
of  surfaces  is  that  of  lines  of  curvature,  which  have  the  property,  that 
along  them  consecutive  normals  to  the  surface  intersect.  Their  dif 
ferential  equation  is  given  most  conveniently  in  the  determinant 
form 


(53) 


drf  —  du  dv  dir 
E  F  G 
D  D'  £>" 


=  o. 


This  differential  equation  is  again  of  the  second  degree,  so  that 
through  each  point  pass  two  lines  of  curvature.  These  are  mutually 
orthogonal,  and  besides  their  directions  are  harmonic  conjugates 
with  respect  to  those  of  the  asymptotic  lines  through  the  same  point, 
as  may  be  seen  readily  from  the  forms 'of  equations  (50),  (52),  and 

(53)- 

Suppose  that  on  a  certain  surface  the  asymptotic  lines  are  known 
to  cut  out  rhombuses.*  This  can  be  expressed  analytically  in  the 
following  way  : 

The  selection  of  the  asymptotic  lines  as  parametric  curves  does 
not  affect  the  appearance  of  the  expression  for  the  element  of  length 
of  arc.  But  since  u  =  const,  and  v  =  const,  must  then  be  the  solutions 
of  (52),  it  follows  that  Z>  =  Z>"  =  o.  Hence  the  differential  equa 
tion  of  the  lines  of  curvature  (53)  reduces  to 

(53')  Edir-  G<h>-  =  o. 

The  elements  of  length  along  the  parametric  curves  are  ^/  E  du  and 
^J  G  dv.  These  will  be  equal  at  every  point  on  the  surface,  and  the 
surface  will  therefore  be  divided  into  rhombuses,  if  V  E  =  A(//,  i')4>(N) 
and  ^/  G  =  X(//,  7')^(7').  (See  corresponding  argument  in  the 
case  of  isothermal  lines  in  Note  III  of  the  Appendix.)  Letting 

*  This  is  known  to  be  the  case  for  surfaces  of  constant  Gauss  curvature,  for 
example. 


82  THEORY   OF   DIFFERENTIAL   EQUATIONS  §25 

j  <£(//)  du  =  U,    I  \l/(i>)  dv  =  F,  the  expression   for  the    element  of 
length  takes  the  form 


The  differential  equation  of  the  lines  of  curvature  takes  the  form 
dU~-dV-  =  Q  ; 

whence  the  equations  of  the  lines  of  curvature  are 

U  +  V  =  const,  and  U  —  V  —  const. 

Since  the  identity  (48)  holds,  it  follows  that  if  the  asymptotic  lines 
divide  a  surface  into  rhombuses,  the  asymptotic  lines  and  lines  of 
curvature  can  be  obtained  by  means  of  quadratures. 


CHAPTER    IV 

DIFFERENTIAL    EQUATIONS    OF    THE    SECOND    AND    HIGHER 

ORDERS 

26.  Twice-extended,  n-times-  extended  Group.  —  A  transformation 
of  the  variables  .v  andjy  carries  with  it  a  transformation  of  the  various 
derivatives  of  y  with  respect  to  x.  Thus,  just  as  the  point  trans 
formation 

*i  =  $(*of)>  ***1fftJ) 

carries  with  it  (§  13) 


dx      By 


so  it  also  implies  d*^ 


dx      dy 
The  transformation 

*!  =  <i>(x, y),    }'i  =  ^(^,y),    y\  =  x(x> y> y)>    }'i"  —  <*(x> y> y'> y") 

affecting  the  four  variables  x,  y,  y',y"  which  is  implied  by  the  point 
transformation  is  known  as  a  twice-extended  point  transformation* 
Starting  with  the  one-parameter  group  of  point  transformations 

( i )  #!  =  <f>(x,  y,  a),  )\  =  if/  (x,  y,  a}, 

*  In  precisely  the  same  way  we  arc  led  to  the  n-times-extended  tr  ana  formation 

—  =  e(x,y,y'ty",  •••,/")). 


84  THEORY   OF   DIFFERENTIAL   EQUATIONS  §26 

by  employing  the  method  of  reasoning  in  §  13,  the  corresponding 
twice-extended  transformations 

(54)  x\  =  <K*>  y>  a)>  }\  =  $(x>  y>  a\  yi  =  V  '  =  x(-v»  }'>  />  <0> 


are  seen  to  constitute  a  one-parameter  group  in  the  four  variables 
x,  y,  y',  y".  This  group  is  known  as  the  twice-extended  group  corre 
sponding  to  (i). 

Writing  as  the  symbol  of  the  infinitesimal  transformation  of  the 
twice-extended  group 

(55) 

where  as  before 


77",  which  is    •  --,  may  be  found  in  exactly  the  same  way  as  ?/  was; 

§a 
thus 


77        SffSp  ^v  ^v-  ^r          dx     dx 


Reasoning  as  before  we  have  the  n-times-exUnded  group 

=  4>(x,  y,  a),  yl  =  $(x,  y,  a,),  yl  =  'j±  =  \(x,  y,  }•',  a), 

"x\ 


§  26      DIFFERENTIAL   EQUATIONS   OF  THB  SECOND   ORDER        85 
the  symbol  of  whose  infinitesimal  transformation  may  be  written 


where 


Remark.  —  While  >/'  is  a  quadratic  polynomial  inj'  ([24],  §  13),  it 
is  seen,  on  expanding  (56), 

(58)         ,"=^v 

djc       5)' 

that  ?;"  is  linear  in  y".     In  the  same  way  ry(*)  is  seen  to  be  linear  in 
y(k)  for  k  >  i,  since 


__ 

By  dy' 


In  I,   Uf,  f  =  o,  r;-i.     /.  17'  =  o,  T;"E: 

«r 

Hence,  U^f=^-- 

dy 

In   II,    Uf=-y-f  +  xd/t    $==-}>,    -n  =  x 

O-^  Cy' 

v"=3/'2  +  4yy",  iiiv=s(2yy" 

Hence, 


in  in,  u/mf-t  ^  =  0,  77=^.    .-.  r  =  v',  17"  =/',  •••»  i?(w)=y"v. 
^^+y:+,"      +  ...+y"». 


86  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§  26,  27 

In  IV,  Uf=x&+y&t  £  =  x,  i,=y.     .-.  ,'  =  o,  i»"=-/',  r/"EE-2/", 


Extend  the  following  groups  : 

Ex.  1.    f  •  Ex.  2.    xf.          Ex.  3.   ^-j 

d,r        .  ox  dx      "  d 


Ex.  4.   a*      +  ^  Ex.  5.    0(.     Ex.  6. 

5jf        J  dy  Jdx 


Ex.  7.  ^. 

dy 

27.  Differential  Equation  of  Second  Order  Invariant  under  a  Given 
Group.  —  The  effect  of  any  transformation  (i)  on  the  variables  x  and 
y  is  to  transform  the  differential  equation 

(60)  /(*,;',/,/')=<>, 

by  the  corresponding  extended  transformation  (54).  In  order  that 
the  equation  (60)  be  invariant  under  the  group  (54),  it  is  necessary 
and  sufficient  that  ([12],  §  n) 


(61)  £/"/=  o  whenever /(*,  y,  y',  y")=o. 

Using  the  same  argument  as  was  employed  in  §  18,  it  is  seen  that 
all  the  differential  equations  of  the  second  order  invariant  under  the 
group  are  obtained  by  equating  to  zero  an  arbitrary  function  of  three 
independent  solutions  of  ([9],  §  n,  footnote) 


§27      DIFFERENTIAL   EQUATIONS  OF  THE   SECOND   ORDER        8/ 

Passing  to  the  corresponding  system  of  ordinary  differential  equa 
tions 

dx  dy  dy1        _  dy1' 


the  first  three  members  are  seen  to  be  the  same  as  those  of  (37), 
§  18.  Hence,  two  of  the  solutions,  u(x,  y)  =  const,  and  u'(x,  }',}>') 
=  const.,  may  be  found  by  the  methods  of  that  section. 

To  find  a  third  solution,  u"(x,  y,  /,  y")=  const.,  which  must  neces 
sarily  involve  y",  use  may  be  made  of  the  two  already  found  to 

eliminate  j>  and  y'  from  -^-  =^-,  or  x  and/  from    -]'    =  —  ,  or  x 
dx       £  dy       TJ 

and  y  from  -^-  =  2_  (whichever  turns  out  to  be  the  simplest).    Each 
dy       77 

of  these  differential  equations  is  linear  since  rj"  is  of  the  first  degree 
in/'  (§  26,  Remark).  This  linear  equation  can  be  solved  by  means 
of  two  quadratures.  (See  El.  Dif.  Eq.  §  1 3). 

Lie  has  given  a  most  ingenious  method  for  finding  a  form  for 
//'i.v,  yf  y,1  y"),  without  any  integration  whatever  when  u  and  u'  are 
known  : 

Consider  the  differential  equation 

(64)  //'(*,  _>',/)-«  «(*,.)')  =  & 

where  a  and  ft  are  constants.  Since  u  and  //'  are  invariants  of  the 
once-extended  group  U'f,  (64)  is  invariant  under  the  group  Uf\  that 
is,  its  integral  curves  are  interchanged  among  themselves  by  the 
transformations  of  this  group.  Keeping  a  fixed,  an  invariant  family 
of  a  single  infinity  of  integral  curves  corresponds  to  each  value  of  ft. 
Still  keeping  a  fixed  and  allowing  ft  to  take  successively  all  possible 
values,  an  infinity  of  such  families,  constituting  a  double  infinity  of 
integral  curves,  is  determined  by  (64).  This  larger  aggregate  is  in 
variant  under  the  group  Uf,  since  each  of  the  constituent  families 
corresponding  to  the  same  value  of  ft  is.  It  is  evidently  the  set  of 


88  THEORY  OF   DIFFERENTIAL   EQUATIONS  §27 

integral  curves  of  the  differential  equation  of  the  second  order  ob 
tained  by  differentiating  (64),  thereby  eliminating  ft  ;  viz. 


du  du<       Bx 


,,. 

(65)  -  --  «  -—  —  o,  or    -—  = 

dx         dx  du 


—  a 


Since  its  integral  curves  are  interchanged  among  themselves  by 
every  transformation  (i),  it  is  invariant  under  the  group  Uf.  Hence, 

by(6l)  fM      \  du' 

Un(  -     —  a  }  =  o  whenever   —  =  a. 
\;///         )  du 

But  a  being  a  constant,  C7"l  — — a  ]=  U"( — •] ;  i.e.  it  is  indepen- 

V  du  J  \dn 

/  J    l\  \          / 

dent  of  «.      £/"[  -  —  )  is  therefore  identically  zero  ;  which  is  sufficient 
,      \d*/ 

to  make  —  an  invariant  of  (54),  ([9],  §  n). 
du 

Since  //'  contains  y'  (§  18),   — ^  o,  and   -       must   contain  y"* 

ay'  du 

Hence,  -—  =  const,  may  be  used  as  the  third  solution  of  (63).     The 

du 

general  solution  of  (62)  may  then  be  written  in  the  form 

(66)  f(u,  «',  ^)  =  o,  or  ^'  =  F(u,  »'). 

\  du  J  an 

This  is  the  general  form  of  the  differential  equation  of  the  second 
order  invariant  under  the  group  Uf.  We  have  therefore  the  follow 
ing  most  important 

THEOREM.  —  If  f(x,  r,  y',  _r")  =  o  is  a  differential  equation  of  the 
second  order  invariant  under  the  group  i'f^  and  if  u  (x,  y)  is  any 

*  An  invariant  of  the  extended  group  U"f  which  involves  j"  is  known  as  a  second 
differential  invariant  of  the  group  Uf. 

t  Attention  should  be  called  to  the  fact  that  while  every  differential  equation  of  the 
first  order  is  invariant  uivk-r  an  indefinite  number  of  groups  (see  \\  15,  17)  a  differen- 


§27      DIFFERENTIAL   EQUATIONS  OF  THE   SECOND   ORDER       89 

invariant  and  //'(x,  v,  v')  is  any  first  differential  invariant  of  Uf,  the 

introduction  of  the  ncu>  rariablcs 

(67)  x=  it  (x,  r),   y  =  u'(x,  y,  y') 

reduces  the  differential  equation  to  the  form 

(66')  '^  =  F(X,y), 

which  is  of  the  first  order. 

In  actual  practice  the  introduction  of  the  new  variables  is  usually 
most  readily  effected  by  noting  that 

Qy  4-  ^  v'  4-  ^  v" 
<ty_dx      d)>          dy'J 


dx  dx      dx    , 

dx      dy " 

is  some  function  of  u  =  x,  u'  =  y,  and  //".  When  this  function  is 
obvious  upon  inspection,  //"  can  be  determined  in  terms  of  x,  y,  '  • 
In  other  cases  it  may  be  necessary  to  solve 

dy       dx      dy  dy' " 

dx      dv 

fory,y',y"  in  terms  of  JT,  y,  '^,  x.     Substituting  these  in  the  differ- 

dx 

ential  equation,  A*  must  disappear,  and  the  resulting  equation  must 
take  the  form  (66'). 

After  having  solved  (66'),  its  solution 
(68)  4>(u,t/',f)  =  o 

a  differential  equation  of  the  first  order.     Rut  owing  to  the  inva- 
riance  of  u  and  u'  (68)  is   invariant  under   Uft  so  that  it  may  be 
)lved  by  the  method  of  §  12  or  that  of  §  20. 

il  equation  of  the  second  (or  higher)  order  is  in  general  not  invariant  under  any 
group.  (See  Note  IV  of  the  Appendix.)  On  the  other  hand,  a  large  number  of  them, 
including  most  of  the  known  forms,  are,  and  these  will  be  considered  in  this  chapter. 


90  THEORY   OF    DIFKKKKNTIAL    EOl'ATIONS  §28 

28.    Illustrations  and  Applications. 

I.    6^"  =  -"  (.     £  =  o,  yj  =  i.    .'.  r)'  =  o,  -r}"  =  o  (§  26).     P^quations 

(6i\   nrp  7  r  7  ri 

\V6J  ^lc  ^v  _  //v  _  <ty  _  ay 

o        i         o         o 


w"  =  y.     Hence,  the  general  type  of  differential 
equation  of  the  second  order  invariant  under  Uf=  •—  is  f(x,  j1',  JIP")  =  o 

0ry"  =  F(xty').     This  equation  is  characterized  by  the  absence  of  r. 
Note.  —  The  transformation  of  variables  x=xt  y  =  }>'  (§  27)  re 
duces  the  differential  equation  to 

(66')  &  =  f(x,y). 

ax 

This  is  precisely  the  usual  method  for  solving  an  equation  of  thus 
type.  (See  El.  Dif.  Etj.  $  57).  Solving  the  solution  of  (66')  for  y, 
it  takes  the  form 


in  which  the  variables  are  separated,  as  must  be  the  case  (I,  §  19), 
since  this  equation  is  invariant  under  the  same  group  (§  27). 

I'.    Uf=  |P£;     It  is  readily  seen  that  the  general  type  of  differen 

tial  equation  of  the  second  order  invariant  under  this  group  is 
/(y,  _y',  y")  —  o,  ffry"  =  F(y,  y').  This  equation  is  characterized  by 
the  absence  of  .Y. 

Note.  —  The  transformation  x=y,  y=y'  (§  27)  reduces  the  dif 
ferential    equation    to    one    of  the    first    order    (66').      Its   solution 

ffy 

y=/(x,  <•),  or  ^  =f(y,  c] 

is  a  differential  equation  with  A*  absent  again,  as  must  be  the  case 
(If,  §  19  and  §  27).  This  is  also  the  usual  method  for  solving  an 
(•([nation  of  this  type.  (See  El.  Dif.  Eq.,  §  58.) 


§28       DIFFERENTIAL    KoUATK  )\S   OF  TIIK   SITCOM)   ORDKK       91 

Remark. — Owing  to  the  simple  form  of  an    equation    invariant 
under   either    of  the    groups  Uf  =  ~  -  or   £^"=  ---,  it   is   frequently 

desirable  to  introduce  canonical  variables  in  case  a  given  differential 
equation  of  the  second  order  is  known  to  be  invariant  under  some 
group.  When  the  introduction  of  canonical  variables  is  not  prac 
ticable,  other  changes  of  variables  reducing  the  group  and  equation 
to  known  forms  may  prove  desirable.  (Compare  §  20.) 


(§  26).     Equations  (63)  are 

(6.3')  *- 

v  —  .vr' 
.-.  u  =  .\-+r,  «'=•    T    ;;/  (§  19)-     Using  the  last  two  members  of 

equations  (63'),  //"  =  — --  — —  •      Hence  the  general  type  of  differential 

^   '  t\f          *i  f 

1  of  tlie  second  order  invariant  under  Uf=  —  V-=    +  .v  ;     is 

v,,,          , -  fo      fy 


Note.  —  The  form  of  this  differential  equation  is  obvious  from 
geometrical  considerations,  since  //  is  the  square  of  the  radius  vector 
to  any  point  on  an  integral  curve,  //'  is  the  tangent  of  the  angle  be- 
twivn  the  radius  vector  and  the  tangent  to  the  curve,  while  u"  is  the 
square  of  the  curvature,  all  of  which  are  left  unaltered  by  the  group  of 
rotations  about  the  origin.  (Compare  §  29.)  In  order  to  integrate 
such  an  equation  the  method  of  I',  Remark,  requiring  the  introduc 
tion  of  canonical  variables  (polar  coordinates  in  this  case)  will  usually 
be  found  desirable. 


Making  use  of  the  fart  that    '  and    •  are  also  first 

VI  +.v'"'  Vi  -r\'''2 

differential  invariants  of  the  group  of  rotations  (II,  Note,  §  19)  other 


92  THEORY   OF   DIFFERENTIAL   EQUATIONS  §28 

possible  forms  of  the  invariant  differential   equation  of  the  second 
order  are 


I  '    \  '  '**  /  '    \ 

~,  -  j  and  --    — ^  =  F\  *-+y2,  — 

ViH-v'v  (I+J'")<          V  Vi  +  y'v 


III.    Uf~y^-     £  =  o,r)=y.     .•.ri'=y')ri"=y"(^26).     Equa- 


_          _ 

o       y       /       /' 


dx      dy 
o       y 

y'  v" 

.'.//  =  jc,  «'  =  '•-,  u"  =  -  —  *     Hence   ///<?  general  type   of  differ  en- 

r\f 

tial  equation    of   the   second   order    invariant  under    Uf  =  y~-    is 


This  equation  is  characterized  by  being  homogeneous  in  y,y',y". 
It  is  evident,  at  once,  that  an  equation  of  this  type  is  left  unaltered 

by  the  affine  group  Uf  =  y-^,  since  the  finite  transformations  of  the 
extended  group  are  x^  —  x,  y\  =  ay,  y^  =  ay1,  y"  =  ay". 

Note.  —  An  interesting  equation  of  this  type  is  the  homogeneous 
(or  abridged)  linear  differential  equation 


The  transformation  x  =  x,  y  =  ~  (§  27)  reduces  the  equation  to 


ax 
a  Riccati  equation.      (Compare  EL  Dif.  Eq.  §  73,  6°). 

*  The  Lie  method  of  vv  27  tjivcs  u"      —       '1'       ,  -'         •'         f-J  ,  and   the  dif- 

<///  r-  v       ^y  ' 

tcrrntial    c(|uation         —('   \   +  /'  M',       ),  \vliich  is,  of  course,  the  same  in  form  as 
that  lound  in  the  body  of  the  text. 


§28      DIFFERENTIAL    F.ni  'A  I  K  ).\S    OK   THE    SE<O.\I)    ORDER       93 


VI.    U/=x      +  ny-.     t  =  x,i,  =  ny.      /.  V  =  («  -  i)/, 
dx  By 

rj"  =  (n  —  2)}'".     Equations  (63)  are 


x       ny      (n—  i)/      (n  —  2)/' 


Hence  the  general  type  of  differential  equation  of  the  second  order 

invariant  under  Uf  =  x  —  -f-  nv  ~—  is  f  ( — ,  — — ,  - —  )  =  o. 
ox       '  dy         \xn    xn~l    xn~-J 

This  equation  is  characterized  by  being  homogeneous  in  x,  y,  y',  y" 
when  these  elements  are  given  the  weights  i,  n,  n  —  i,  n  —  2  respec 
tively. 

Note.  —  Boole  called  an  equation  of  this  type  homogeneous,  and 
gave  as  a  method  for  solving  it  the  transformation  x  =  log  x,  y  =  ^-. 

(See  Boole,  Treatise  on  Differential  Equations,  p.  215  ;  Forsyth, 
Treatise  on  Differential  Equations,  §  55).  The  new  variables  in  this 
transformation  are  a  set  of  canonical  variables.  (Compare  I',  Remark.) 

III'.    Uf=.\—  is  a  special  case  of  VI.     Here    ;/ =  o,   and  the 
ox 

invariant  differential  equation  is  of  the  form  f(y,  xy',  x1)'")  =  o. 

This  equation  is  homogeneous  in  x.  y',  y"  when  these  elements 
have  the  weights  i,  —  i,  —  2  respectively  ;  the  weight  of  y  being  zero, 
the  manner  in  which  this  variable  enters  plays  no  role. 

III.  Uf=L\^-  may  also  be  looked  upon  as  a  special  case  of  VI, 

corresponding  to  the  value  ;/  =  oo.  Boole  deduced  a  special  method 
for  this  case  (see  Boole,  p.  220;  Forsyth,  §  55)  which  is  exactly  that 
of  §  27  for  this  case. 

IV.  Uf=x  —  -\-  v---  is  the  special  case  of  VI  for  n  —  i.     The 

d\      -  dy  ( 

invariant  differential  equation  is  of  the  form  f(^  ,  r',  .vv")  =  o. 

\  oc 


94  TIIKORV    OF    niFFKKKNTIAI.    KOUATK  >\S  §28 

V.    Uf=x  <£*  —  y  —  is  the  special  case  of  VI  for  n  —  —  i.     The 

dx         dy 

invariant  differential  equation  is  of  the  form  f(xy,  x'-y',  .ry")  =  o. 

vii.  u/=  *(*)§£•    t  =  o,  -n  = 

It  is  readily  seen  that 


Hence  ///<?  general  type  of  differential  equation  of  the  second  order 
-^- 


invariant  under  Uf=$(x}-^-  is  f(x,  <j>y'  —  $'}',  <f>y"  —  <j>"y}  =  o,  or 


Note.  —  An  interesting  equation  of  this  type  is  the  complete  linear 
equation 

(70)  /'  -f  />(*)/  +  Q^)y  =  X(x), 

which  is  obtained  from  the  general  form  by  letting  F  be  linear  in 
7/EE  <£/  —  <£'v.  Bearing  this  fact  in  mind,  it  is  clear  that  y  =  <£(.v) 
satisfies  the  abridged  equation  (69),  obtained  from  (70)  by  replacing 
X(x)  by  o.  Conversely  it  is  readily  seen  (and  will  be  left  as  an  exercise 
to  prove)  that  \fy=y0  is  a  solution  of  (69),  (70)  is  invariant  under  the 

group  Uf=\\)  -.  The  transformation  x  =  x,  y=y{ty'  —  yjy  (§  27) 
reduces  the  equation  to  the  linear  equation  of  the  first  order 

(71)  ' 


This  property  of  the  complete  linear  differential  equation  of  the 
second  order  of  reducing  to  one  of  the  first  order  by  a  transforma 
tion  that  is  known  when  a  particular  integral  of  the  corresponding 
abridged  linear  equation  is  known  is  not  new.  (See  7v7.  Dif.  Rq. 
§  53,  i°.)  The  transformation  employed  above  yields  an  equation 


§28      DII'KKRKNTIAL    EQUATIONS    OF  TIIK   SKCOND   ORDER       95 

bearing  a  more  striking  resemblance  to  the  original  equation  than 
the  transformation, 


usually  employed.     The  new  variables  in  this  transformation  are  a  set 
of  canonical  variables  (I',  Remark). 

Other  groups   whose   invariant    differential    equations  are  readily 
found  are  the  following  : 


VIII.      tf/EEitfy)/-.  f(x/-S-r  -•>-?-     =0. 


VIII'.    Uf=  *(*)  f(y,  </>y,  4r>"  +  Wy')  =  o. 


X.    Uf= 


XII.    Uf=a      +  l>. 

In  Table  II  of  the  Appendix  will  be  found  a  list  of  the  more  com 
monly  occurring  and  readily  recognizable  classes  of  equations  of 
higher  order  than  the  first  invariant  under  known  groups. 

Ex.  i .  xyy11 4-  -*y2  —yy'  —  o. 

This  equation  is  invariant  under  the  group  IV:    £^"=  x~-\- y-fr* 
Introducing  the  new  variables 

*='[,>  y=y' 


96  THEORY    OF    DIFFERENTIAL    EQUATIONS  §28 

the  equation  takes  the  simple  form 

^+?  =  o. 

dx        X 

vv' 

Integrating  xy  =  a,  or  •*—  =  a. 

oc 

Integrating  again  ax1  —f  =  b. 

Note.  —  Inspection  shows  that  this  equation  is  also  invariant  under 

III:    Uf=y^t  and  III':    Uf  =  x-t> 
dy  ox 

Ex.  2.  (.r-  +f-)y"  +  2  (y  -  */)(i  +  /2)  =  o. 

This  equation  is  invariant  under  \he  group  II  : 


Introducing  the  canonical  variables  (in  this  case,  polar  coordinates) 


the  equation  takes  the  form  —  =  -f-  y  =  o. 

tbr 

Here  the  independent  variable  is  absent,  but,  instead  of  using  the 
•method  indicated  by  the  general  method  of  §  27,  it  will  be  simpler 
to  solve  this  linear  equation  with  constant  coefficients  by  the  usual 
method  for  such  an  equation.  (See  EL  Dif.  Eq.  §  45.) 

y  =  a  cos  x  -f  b  sin  x, 
To  pass  back  to  the  original  variables,  multiply  by  y,  whence 


§§28,29    DIFFERENTIAL   EQUATIONS  OF  THE  SECOND  ORDER     97 

Note.  —  This  differential  equation  is  also  invariant  under  IV. 
Ex.3,     x2}'}'" —  (xy'—}')-  =  o.     (Invariant  under  III,  IV,  .   .   .   .) 
Ex.  4.     ji'y"  +  to''  -  J')2  =  o.        Ex.  5.     x'2y"  =  xy'  -  y. 
Other  equations  invariant  under  known  groups   appear   in  §§39 
and  40. 

29.  Further  Applications.  —  Besides  being  able  to  recognize  a 
-roup  under  which  a  given  differential  equation  is  invariant  from  the 
characteristic  properties  given  in  §  28 
and  enumerated  in  Table  II  of  the 
Appendix,  it  is  possible  at  times,  to 
find  such  a  group  from  the  nature  of 
the  problem  giving  rise 
to  the  differential  equa 
tion.  As  examples,  the 
following  may  be  noted  : 
i°  The  group  of  rota 
tions  about  the  origin 
zs  r\s 

leaves  unaltered 


dx         dy 

R  =  the  radius  of  curvature  of  a  curve  at  any  point, 
p  =  the  radius  vector  to  any  point  on  the  curve, 
r  =  the  radius  vector  to  the  centre  of  curvature, 

the  distance  from  the  origin  to  any  line  (such  as  the  tangent 
or  normal)  connected  with  the  curve,  thus  OM  and  ON, 
/W=the  polar  subtangent,  =  ON, 
PN=  the  polar  subnormal,  =  OM, 

i/'  =  the  angle  between  the  radius  vector  and  the  tangent, 
the  remaining  angles  of  the  triangle  OCP. 

Hence  a  family  of  curves  defined  by  a  relation  between  any  or  all 
of  these  is  unaltered  by  this  group ;  the  differential  equation  of  the 


98  TIIFORY    OF    D1FFFRFXTIAL    FOUATIONS  §29 

family  is  therefore  invariant  under  it.  Passing  to  polar  coordinates 
(the  canonical  variables)  will  usually  be  found  desirable  in  this  case. 

2°  The  similitudinous  group  Uf^x  —  -f- y  —  leaves  unaltered 

civ         dy 

0  =  the  angle  between  the  initial  line  and  the  radius  vector, 

T  =  the  angle  between  the  initial  line  and  the  tangent  to  the  curve, 

<£='the  angle  between  the  initial  line  and  the  radius  vector  to  the 
centre  of  curvature, 

<//  =  the  angle  between  the  radius  vector  and  the  tangent  to  the 

curve, 

the  ratio  of  certain  lines  connected  with  the  curve,  such  as 
radius  vector,  radius  of  curvature,  radius  vector  to  the  centre  of  curva 
ture,  intercepts  of  the  tangent,  normal,  or  of  the  curve  itself,  sub- 
tangent,  subnormal,  length  of  tangent  or  normal  from  a  point  on  the 
curve  to  one  of  the  axes,  and  the  like. 

Hence  this  group  leaves  unaltered  the  differential  equation  of  a 
family  of  curves  defined  by  a  relation  between  any  of  the  above  in 
variant  configurations.  Passing  to  canonical  variables,  or  to  polar 
coordinates  (thereby  reducing  the  group  to  III')  may  simplify  the 
problem  of  solving  the  differential  equation. 

3°  Certain  configurations  could  be  enumerated  as  invariant  under 

the  groups  of  translations  Uf  =  ~  and  Ufm4~*     But  as  in  either 

case  one  of  the  variables  is  absent  in  the  resulting  differential  equa 
tion,  the  latter  will  suggest  the  group  without  considering  the  defini 
tion  of  the  integral  curves. 

Ex.  Kind  the  family  of  curves  for  which  the  radius  vector  to  any 
point  of  a  curve  is  perpendicular  to  the  radius  vector  drawn  to  the 
centre  of  curvature  of  the  curve  at  that  point. 

The  differential  equation  of  this  family  must  be  invariant  under 
the  group  of  rotations  11  and  also  the  similitudinous  group  IV. 


§§29,30         DIFFERENTIAL  KOUATK  >\S  OK  IIKlIlF.k  OkDKR         99 
Noting  in  Fig.  4  that  the  triangle  POC  is  right-angled  at  O, 


=  cos,  or       =  s 


Hete  P  =  Vxf+'f,   R  =  ,   tan  «//  =—  :      -     Hence   the 

/'  *  +yy 

differential  equation  is 

(.v2  +/)/'-  (i  +y*)(y-xf)  =  °- 

30.  Differential  Equation  of  Order  Higher  than  the  Second  Invari 
ant  under  a  Given  Group.  —  The  method  of  §  27  can  be  extended 
without  change  to  differential  equations  of  higher  order  : 

A  differential  equation  of  the  ;/th  order 

(72)  /(*,;',/,/',  •••>j("))=° 

is  invariant  under  the  group  Uft  if  and  only  if 

(73)  U(n)f=o  whenever/^  o. 

All  the  differential  equations  of  the  «th  order  invariant  under  the 
group  are  obtained  by  equating  to  zero  an  arbitrary  function  of  ;/  -f-  i 
independent  solutions  of 


These   independent   solutions   may  be   obtained    from   the  corre 
sponding  system  of  ordinary  equations 

dx      f/v_,/r'      </v"_  <&*> 

\15)  T  —   '—     ,  —    '  TT  —  "'  =  -'~rrr* 


It   was    seen    in   §  27   that   if   //(.v,  v)   is  an  invariant  of   (Y,  and 
i/'(.\\  v,  v')  is  a  first  differential  invariant,  then  '       is  a  second  differ- 

(/!/ 

ential  invariant.      Hence, 

(76)  *£-<&*  ft 


100  THEORY   OF   DIFFERENTIAL    EQUATIONS  §30 

is  an  invariant  differential  equation  of  the  second  order  for  all  values 
of  the  constants  a  and  ft.  Its  integral  curves  constitute  an  invariant 
family  of  oo2  curves.  The  <x>1  differential  equations  of  the  second 
order  obtained  by  keeping  a  fixed  and  giving  to  ft  all  possible  values 
have  for  integral  curves  oo1  such  invariant  families  of  oo-  curves. 
Grouping  all  these  curves  into  one  aggregate  of  <x>3  curves,  this  aggre 
gate  is  invariant  under  the  group  since  each  of  the  families  is.  The 
differential  equation  of  this  family  is,  therefore,  invariant.  It  is  ob 
tained  by  differentiating  (76),  thus  eliminating  ft, 

(     \  —(^\—    ^_  d~11'        _ 

{77)  dx\du)     adx~  ~dtf 

In  order  that  (77)  be  invariant,  we  must  have  from  (73) 

TJ.,.(d"u{        \  ,  (/-//' 

U  —  a  •=  o,  whenever =  a. 

\  dir          J  dir 

"Rnf-  /"/'" 

u      i      ,  ./        lv  I  —  ^      ~T~9    ' 
dir          J  dir 

i.e.  it  is  independent  of  a.     Hence,  if  (77)  is  to  be  invariant,  £/"' — — 

72         I  **** 

must  vanish  identically.     So  that  -    -  is  a  solution  of  (74).     Since  it 

atr 

contains y  (as  may  be  seen  readily),  it  is  independent  of  //,  //',  — « 

dlt 

In  the  same  way  it  can  be  shown,  step  by  step,  that  a  set  of  inde 
pendent  solutions  of  (74)  is 

u'  du-   <w        </n-v. 

du       dir 


Hence  the  general  type  of  differential  equation  of  order  ft  invariant 
under  the  group  Uf  is 


§30  DIFFERENTIAL    EQUATIONS   OF    IIKUIER   ORDER  IOI 

We  have  then  as  an  extension  of  the  theorem  of  §  27  the  following 

THEOREM.  —  If  f(x>  }',  _v',  y",  •-,  y(H))  =  o  ls  invariant  under  the 
gn>itf>  L'f,  ami  ////(.v,  r)  /v  an\  invariant,  and  //'(.v,  v,  r')  is  any  first 
differential  invariant  of  Uf,  t/ie  introduction  of  the  new  variables 

(79)  *  =  M(-V>  y\  y  =  "'  (x>  y>  y') 

reduces  the  differentia]  equation  to 


u>)iich  is  of  order  n  —  i. 

After  having  integrated  (78'),  its  solution 


is  a  differential  equation,  also  invariant  under  Uf,  since  u  and  //'  are. 
Hence  it  may  be  solved  by  the  method  of  §  1  2  or  of  §  20. 

Many  of  the  arguments  of  §  28  can  be  used  here,  almost  without  a 
single  change.  Consequently,  the  results  only  will  be  given,  it  being 
left  as  'an  exercise  for  the  student  to  fill  in  the  steps. 

I.      The  general  type    of  differential  equation    of  the    nth   order 

invariant  under  Uf=~~  is/(x,  y',  y",  •••,/"))=o,  which  is  charac 

terized  by  the  absence  of  y. 

The  transformation  y=y't  reducing  the  differential  equation  to  one 
of  order  n  —  i  constitutes  the  usual  method  for  solving  an  equation 
of  this  type.  (El.  Dif.  Eq.  §  57.) 

I'.      The  general  type   of  differential  equation    of  the   nth  order 

invariant  under  Uf=^  is  /(y,  v',  v",  •••,  v(n))=o,  which  is  charac 
terized  by  the  absence  of  jt\ 

The  transformation  x  =_v,  y  =  y',  reducing  the  differential  equation 
to  one  of  order  ;/  —  i  constitutes  the  usual  method  ior  solving  an 
equation  of  this  type.  (El.  Dif.  Eq.  §  58.) 


IO2  TIIKOKY    OF    DIFFERENTIAL    EQUATIONS  §30 

The  remark  of  I',  §  28  with  reference  to  the  introduction  of 
canonical  or  other  variables  when  a  group  is  known  under  which 
a  given  differential  equation  is  invariant  applies  equally  well  here. 

I.      The  general  tvpc  of  differential  equation  of  tJie  ntJi  order 
invariant   under    Uf  =  y—    is  f(  x,  --,"—,  •••,  •-  -  -  ]  =  o,    which    is 

' 


y   y 

characterized  by  being  homogeneous  in  y,  v',  v",  ~-,y(n). 

VI.      The  genera!  type  of  differential  equation  of  tJie  ntJi   order 

<V  ,       V  •    f(y       i"        r"  v(H}  \ 

invariant  under  C/ES.V/    -f-  r\—  -  is  J\  ~    ,     — ,,  — — ;,>  •••?"  =  o, 

c).v       "  By         \.v''    .v''        .vr  xr     ) 

which  is  characterized  by  being  homogeneous  in  x,  y,  y',y",  -',y("\ 
when  these  elements  are  given  the  weights  i,  r,  r—i,  r—2,  ••«, 
r—  n  respectively. 

As  special  cases  of  this  group  may  be  mentioned 

IV:   r=i, 

V:   r=-i,  /(AT,  .vV,  W,  -.,  .v"* V"0=  o, 
III'  :   r=o,  /( y,  xy\  ^;r",  ••-,  .v"_v'"))=o, 

III  :   r=&2.     The  invariant  equation  in  this  case  is  more  readily 
recognized  by  the  other  characterization  given  under  111  above. 

VII.      The  general  type  of  differential  equation  of  the  ntli  order 
invariant  uihler 


or       ^,»  -  <";>  =      .v,  ^    -  <£,  ^,r    -  <^>,  •  •-,       n~    -      -?. 

Note.  —  An  interesting  equation  of  this  type  is  the  complete  linear 
equation 

(80)  y>  +  I\y( " -1'  +  7'8y »-2>  -f  ...  '-h  /^O'"  +  /^,-ui''  4-  /'«.v  -  A^. 


j  30  niKKKKKXTIAL    EQUATIONS    OK    MIOHKR    ORDER  IO3 

If  T=  ]<o  is  a  particular  solution  of  the  abridged  equation  obtained 
by  replacing  X  by  zero,  (80)  is  invariant  under  Uf=yn  =-  • 

The  transformation  (79)  y=y0y'  —}'0'y  (or  y=yHy,  resulting  from 
the  introduction  of  canonical  variables)  reduces  (80)  to  a  linear  equa 
tion  of  order  //  —  i  (fi/.  Dif.  Eq.  §  59),  but  the  resemblance  of  the 
resulting  equation  to  the  original  one  is  not  as  striking  as  in  the  case 
of  the  linear  equation  of  the  second  order  (VII,  Note,  §  28). 

XII.    77ir  gene  i\i  I  type   of  differential  equation   of  the   nlJi   order 

invariant  under  rf=a^+l&  is  f(bx  -  ay,  v',  r",  •••,  vril))=o. 
d.\-         dy 


CHAPTER   V 

LINEAR  PARTIAL  DIFFERENTIAL   EQUATIONS   OF   THE 
FIRST   ORDER 

31.    Complete  System.* 

THEOREM  I.  —  If  <£(.v,  v,  z)  is  a  solution  of  the  two  independent  'f 
linear  homogeneous  equations 


=Pi(x,  y,  z) 


B/U*,*  z)      +&(*,*  z)      +*,{*>*  z}      =  0, 


it  is  a/so  a  solution  of 


4-  (A,  Q,  - 


where  (A^A^)  is  the  alternant  of  the  operators  A^  and  A.,  (§  14). 
For  (AiA2)<f>  =  A^A^)  —  A.2(Al(f>)  =  o,  since  A^  =o  and  .  /L,<^  =  o. 


*  Only  so  much  of  the  theory  of  complete  systems  and  only  such  methods  for  their 
solution  as  seem  necessary  lor  our  immediate  purpose  are  given  here,  tor  an  excel 
lent  detailed  treatment  of  this  subject  the  student  is  referred  to  Goursat-Bourlet,  Inte 
gration  i/t's  equations  au\  derive  es  partielles  du  premier  ordre. 

t  r  equations  of  this  type  in  n  variables  are  said  to  be  independent  if  it  is  impossible 
to  find  r  functions  a\,  ff^>,  •••,  <rr  of  the  variables  such  that 


In  the  case  of  r  =  2,  this  amounts  to  saving  that  the  equations  are  independent  if 
one  of  them  is  not  a  multiple  of  the  other. 

104 


§3i  LINKAR    I'AKTIAL    I  >I  !•  !•  I-.UKNTI  AL    EQUATIONS  105 

If  three    linear    equations    in    three    variables    A}f=  o,  A.j,f  =  o, 
A»f=o  have  a  common  solution  <f>(x,  y,  z]  other  than  a  constant, 

dd>    dd>    dd> 

_r  satisfy  the  three  homogeneous  linear  relations 
dx    dy      dz 


Since  <f>(x,  y,  z)   is  not  a  constant,       -,    -     ,  cannot  all  be 

identically  zero.     Hence 

i  a  *i 


a  ^i 

It  follows  that  three  functions  <ri(x,y,  z),  <r.2(x,  y,  z\  (rs(x,  y,  z)  can 
be  found  *  such  that 
(8  1  )  o-i^i/4-  <r2^2/  -f-  vsAJ  =  o  ; 

i.e.  the  three  equations  are  not  independent.     Hence  follows 

THEOREM  II.  If  the  three  equations  in  three  variables  A\f=  o, 
A.>f=  o,  ^/;i/=  o  have  a  common  solution,  other  tlian  a  constant, 
they  are  not  independent;  or  stated  otherwise,  three  independent 
linear  homogeneous  partial  differential  equations  in  tJiree  variables 
cannot  liave  a  common  solution,  other  tJian  a  constant. 

From  Theorems  I  and  II,  it  follows  at  once  that  if  A^f=  o  and 
A.,f=  o  have  a  common  solution, 

(82)  (^M,)/EE  Pl(x,  yt  ZJAJ+  p,(.v,  r,  •  ,.  /._  /; 


*  Thus,  for  example,  one  may  take  for  <r\.  o-._>,  <T;}  any  tliree  functions  proportional 

to  the  col.u-tors  of  the  i-oricsMoudini;  dements  ot  any  column  m  A. 


106  THEORY   OF   DIFFERENTIAL   EQUATIONS  §31 

Clebsch  gave  the  name  of  complete  system  to  a  pair  of  independent 
equations  A^f  —  o  and  A.2f=  o,  which  are  connected  by  the  rela 
tion  (82).  The  last  statement  may  therefore  be  put  into  the  form 

THEOREM  III.  If  Alf=o  and  A  f  •=.  o  have  a  common  solution, 
they  form  a  complete  system. 

Conversely,  we  shall  prove  the  very  important 
THEOREM  IV.    If  ^/=  o  and  A.1f  =  o  form  a  complete  system, 
they  have  a  common  solution. 

In  order  to  do  this  it  is  necessary  to  prove  two  lemmas. 

LEMMA  I.  If  A^f  —  o  and  A.,f  =  o  form  a  complete  system,  any 
pair  of  equations  formed  of  independent  linear  combinations  of  these, 
also  form  a  complete  system. 

The  equations 


(8  ) 

=    2,  y,  z)        +       x,  y,  z).>=  o 


are  independent  if  \^2  —  ^i  ^  °-     Then  y^/and  A.J  can  be  found 
as  linear  functions  of  A^f  and  >l>/from  (83). 

Since  A-^J  '=  o  and  A<2f—  o  are  supposed  to  form  a  complete  system, 


is  seen  to  be  a  linear  function  of  A^f  and  A*f,  and  therefore  of 
>f,/and  A-if,  which  proves  the  lemma. 

Moreover,  any  common  solution  of  sl}f=  o  and  A.tf=  o  must  be 
such  for  >0j/==o  and  yf^/^o,  and  vice  versa.  Hence  the  two  systems 
are  said  to  be  equivalent,  or  each  is  said  to  be  equivalent  to  the  other. 

A  system  equivalent  to  the  original  system  is  obtained  if  the  equa 
tions  of  the  latter  are  solved  for  two  of  the  three  partial  derivatives 

-^-,   /  ,   ;   •     This  can  always  be  done,  since  all  three  of  the  deter- 

- 


minants  in  the  matrix 


y> 


Q, 


§3i  LINEAR    PARTIAL   DIFFERENTIAL   EQUATIONS  IO/ 

do  not  vanish  identically,  AJ=  o  and  A.,f=  o  being  independent 
equations.     If,    in    particular    J\Q,  -  1>,Q,  =  D  =£  o,   the    equations 

may  be  solved  for  -j-  and  ^ ,  thus  giving 


where    Rl=.     *,  =  =:.       Here   *,=-§, 

/xj  =  -  ^  ,    X,  =  -  ^  ,    n»=-\  and    X^o  -  X,/^  =  -^  =£  o,  since  all 

functions  involved  are  supposed   to   be    generally  analytic.     Hence 
equations  (83')  are  independent.     This   fact  is  also   obvious   upon 

inspection,  since  the  first  equation  is  free  of  v~,  while  the    second 

/)/ 
does  not  contain  ~  •     Moreover 

dx 

(84)  (A^f^o. 

For,  since  AI/=  o  and  A>if—  o  form  a  complete  system 

(82)  .     (Aj.yEEp 

In  the  case  of  equations  (83') 


which  is  free  of  both   /-   and  -  -  .    Hence  pi  and  p.,  in  (82)  must  both 
dx  dy 

be  zero,  and  the  form  (84)  follows. 

A  complete  system  for  which  pl  =  p2  =  o  is  called  a  Jacobian*  com 
plete  system.  We  have  thus  established 

*  Originally  this  term  was  applied  only  to  a  complete  system  in  the  special  form 
(83').  Lie  and  other  mathematicians,  however,  used  it,  as  above,  to  apply  to  the  more 
general  class  of  complete  systems;  (SIT  Lie,  DifertntialgUifktaig€*t  p.  202;  Goursat- 
BourU-t,  /<>6-.  cit.,  p.  347;  also  Encykkptidie  dcr  M^t/icinatisi/icn  U'lssi-HSi/iaftctt,  Hanil 
HI.  P-  315). 


108  THEORY   OF   DIFFERENTIAL   EQUATIONS  §31 

LEMMA  II.  —  A  Jacobian   Complete    svstein    can    always  be  found 

equivalent  /<;  a  gircu   complete  s\  stein. 

Remark.  —  It  should  be  noted  that  this  equivalent  Jacobian  sys 
tem  is  not  unique,  since  starting  with  such  a  one,  the  system  obtained 
by  taking  any  pair  of  independent  linear  combinations  of  these  equa 
tions  with  constant  multipliers  is  another  system  of  the  same  sort. 

It  is  easy  to  show  that  a  Jacobian  complete  system  has  a  solution. 
Suppose  that  A-^f  '  =  o  and  A»/=  o  form  such  a  system.  Then 

(84)  (*A)/=  Ai(A'./)-A*(Ai/)=  o. 

If  u(x,  y,  z)  and  v(x,  r,  2)  are  two  independent  solutions  of  one  of 
the  equations,  say  Alf=o,  any  function  of  //  and  v  will  equally  well 
satisfy  this  equation.  \\.  remains  to  find  such  a  function  of  them, 
F(tt,  v),  that  it  shall  also  be  a  solution  of  the  other  equation  >f2/—  o  ; 
that  is, 

(85)  .  A*F(u,  v)s  °f  A,u  +  ~A,v  =  o. 

()//  VV 

Replacing  /  in  (84)  by  u  and  v  successively, 

=  o  and  A^A.v—  AJiA^=  o. 


Since  A^u  =  o  and  A\v  =  o,  it  follows  that 

Al(A.>!t)=o  and  Ai(A%v)=Q. 

Hence  Azu  and  Av  are  functions  of  //  and  v,  say  $(//,  v)  and  ^(//,  i>) 
respectively,  and  the  equation  (85)  to  determine  F(u,  v)  is 

(85')  <K",  7.')^+  t(»,V)d£=0. 

The  solution  of  this  equation  (which  is  known  to  exist  by  the  gen 
eral  existence  theorem)  is  ;i  solution  of  the  Jacobian  system  A}/=o, 
A-J=  o,  and  consequently  of  the  equivalent  complete  system  At/=  o 
and  A^f=  o.  Theorem  IV  is  thus  proved. 


LINEAR    PARTIAL    DIFFERENTIAL    EQUATIONS 


109 


All  that  has  gone  before  can  be  extended  at  once  to  homogeneous  linear  equa 
tions  in  //  variables. 

Without  changing  a  word  in  the  proof  of  Theorem  I  we  have:  If 
0  (•*'!>  -*"2>  •••»  •*«)  -is  a  solution  of  the  two  equations 


+  /'•>„  (>i,  .*••>,  •••,  .vw)  ,f  -  =  o, 
<•/»« 

zY  is  also  a  solution  of  (A\A>>}f  =  o. 

As  before,  ?/"  w  equations  hare  a  common  solution,  other  than  a  constant,  the 
equations  cannot  t>e  independent.     For  the  determinant  of  the  coefficients 


must  vanish.     Hence  a  relation  of  the  form 

0V/  1/  +  «r.j//2/+  ••• 
must  exist. 

Starting  with  /•  independent  eqtUttioiU 
.-/i/=o,    A.2/=o, 


Arf=o(2<r<n) 


with  a  common  solution,  all  the  equations 

(/MJ/zrO,      (*V«=    I,   2,   3,    ...,r), 

will  also  have  this  solution.  Some  or  oil  of  th.-se  equations  may  be  independent 
of  the  original  equations.  Adjoining  these  to  the  latter,  the  proems  may  In- 
repeated  as  long  as  independent  equation!  can  be  lound.  This  process  must 
come  to  an  t  nd  before  the  total  number  of  equations  reaches  //.  For  it  ha-;  ju-t 
been  seen  that  there  cannot  be  //  independent  linear  equations  in  ;/  variables 


1  10  THEORY  OF  DIFFERENTIAL   EQUATION'S  §31 

having  a  common  solution,  other  than  a  constant.     We  have  thus  obtained  a 
system  of  s  equations 

Aif=o,   A-2/=o,    -•-,   A.f—  o  (r  <  s  <  ;/) 

such  that  (AtAlty=piAi/+  p-j.-/,>/+  ...  +  p,A,f, 

(/,   K=l,2,   3,    .-.,.*), 

Such  a  system  constitutes  a  complete  system.  We  have  thus  shown  that  if  '  r 
equations  hare  a  common  solution,  everv  member  of  the  complete  system  determined 
by  them  has  that  solution. 

It  will  be  left  as  an  exercise  for  the  student  to  show  that  starting  with  any 
complete  system  an  equivalent  Jacobian  system*  can  be  found.  The  method  is 
identical  with  that  given  above  for  three  variables. 

That  a  Jacobian  complete  system  (and,  therefore,  any  complete  system)  of  s 
equations  in  ;/  variables  has  n  —  s  independent  solutions  may  be  proved  in  a 
manner  entirely  analogous  to  that  used  above  for  s  =  2,  n  =  3.  To  illustrate,  the 
case  for  s  —  3,  n  =  5  will  be  given  without  detail  : 

The  equation 


has  four  independent  solutions  m,  #•_>,  «;,  «t  (F<t-  Dif.  Rq.  §  79).  The  problem 
is  now  to  show  that  some  function  J'\u\,  //-j,  w;5,  //4)  of  these  will  satisfy  both 
Arf  —  o  and  A%f  =  o. 

Since  ut  for  i  —  I,  2,  3,  4  satisfies  A\f  —  o,  it  follows  on  replacing  /  by  ;/,  in 
the  identity 


that  A»iii  is  also  a  solution  of  A\f—  o.  Hence  A»Ui  must  be  some  function  of 
«i»  "2,  »3f  «4,  say  0<(«i,  //_•,  //;•„  //.,),  for  /  =  i,  2,  3,  4.  If  /'Ms  any  solution  of  the 
equation  involving  the  four  variables  MJ,  //._.,  n-\t  u*, 


it  will  be  a  solution  of  A\f—  o  and  ./•_>/=  o. 

*  A  Jacobian  complete  system  of  J  equations  is  one  for  which 

(,/,./J/   :0(*,*-I,a,3,...,x). 

See  prt^vious  footnote. 


§§.31,32      LINEAR    1'AKTIAL   DIFFERENTIAL   EQUATIONS  III 

This  equation  has  three  independent  solutions  v\,  v^  v-^.  Any  funetion  of  these 
will  be  a  solution  of  A\f—  o  and  A*/  —  o  ;  and  conversely,  every  solution  com 
mon  to  A\f—Q  and  A-^f—Q  must  be  a  funetion  of  7'j,  z*>,  v^.  It  remains  to 
show  that  some  function,  4>(z>i,  z>2,  ^3),  of  them  will  satisfy  A^f—  o. 

As  before,  it  follows  on  replacing/  by  ?',-  in  the  identities 


that  ^;;^j  is  a  solution  of  both  A\f—  o  and  A-if  —  o.  Hence  .•/;{7't-  must  be  some 
function  of  z/[,  ZA-,  ?',;,  say  ^,(^1,  ?/.-,  ?';;),  for  *  =  I,  2,  3.  The  function  4>  may  then 
be  any  solution  of  the  equation 


This  is  known  to  have  two  independent  solutions.  Each  of  these  is  the-.efore  a 
solution  of  the  complete  system,  and  there  can  be  no  others. 

32.  Method  of  Solution  of  Complete  System.  —  To  actually  find 
the  solution  common  to  the  members  of  a  complete  system  A^f=  o 
and  ^-2/—  °  ^  ^  not  necessary  to  pass  to  an  equivalent  Jacobian 
system.  If  it  and  v  are  two  independent  solutions  of  one  of  the 
equations,  A^f  —  o,  it  is  known  that  some  function  F(u,  v)  is  a  solu 

tion  of  the  other  ;  i.e. 

dF  A/? 

(85)  A2F(u,  v)  =  A.u-g  +  A&  ~  =  o. 

or 

(86)  f 


Knowing  that  some  form  of  *F(tt,  v)  must   satisfy  this   equation, 

whence  —  -  and  -   -  are  also  functions  of  u  and  v,  —  ^    must  be  a 

OK  dv  A«u 

function  of  //  and  v.*  Hence  (86)  may  be  written  as  an  equation  in 
these  two  variables  only,  and  the  usual  method  of  solution  for  such 
an  cc  juation  may  then  be  followed. 

*  It  should  be  noted  that  in  this  ease,  unlike  in  the  case  of  a  Jacobian  complete  sys 
tem,  A.M  and  A.v  need  not  be  Junctions  ot  u  and  v,  although  they  may  be. 


112  THEORY   OF  DIFFERENTIAL   EQUATIONS  §32 


Since  (_AiAJ)f=Aif,  these  form  a  complete  system. 
Here  //=jr,  v  =  z  are  solutions  of  A^f=  o.     Then  A»t(=v  —  itt 
A»v  =  z  =  v,  and  equation  (85)  may  be  used  to  determine  F\  thus 

rV''  ,       QF 

v-±-  +v--  -=o. 

O//  027 

The  general  solution  of  this  is  any  function  of  -  .     Hence  the  com- 

u 
mon  solution  of  the  complete  system  is  any  function  of  -. 

x  v  y 

Or  starting  with  u  =  -,  v  =  "  ,  the  solutions  of^.,/=o,and  noting 

y          z 

that  A&  =  -  ,  Af)  =  -  ,  whence   -^  =-^  =  -,  equation  (86)  is 
>'  2  A&      z      u 

dF  ,  7-  5^ 
--  1  ---  =  o. 
au      1  1  dv 

Its  solution  is  -=-v,  giving  the  common  solution  of  the  system  01 
equations. 

Ex.2. 


,~(x-  +  r  +.v-       +      +  r-  «f-hjw    -=  o. 
These  form  a  complete  system,  since  (AlA.?)/=  A^f. 


}.       2u  =  ^-          Z- 

*  2 

-xz-x 


»  =  -,#s-    are  solutions  of  A}/=o. 

* 


=  --  =  _       and  equation  (86)  is 

r  7^ 


Its  solution  is  //-'  -4-  ir.      Hence  the  common  solution  of  the  com 
plete  system  is  any  function  of  -  ~*-  . 


§§32.33      I.INKAK    I'AKTIAI.    I  IIFFEKKNTIAI,   EQUATIONS  113 

Ex.  3.    AJ-r     -  *=  o,  A,f=       =  o. 


Ex.4.    JJ-x       +  z 


Ex.5.    W*V  +  t£  +  'tf-i    A^X!i{  +,f  +  sf=o. 
ojc        a          dz  ox         d          dz 


Ex.  6.    AJ=  (A-  -.v  +  8)      -  2j      -f-  (,v  -  y  +  f)=  o,. 


Ex.  7.    ^/^  (»  -  j)       4-  (yz  -  A-)+  (i  -  ^)=  o, 


-  o. 


33:  Second  Method  of  Solution.  —  If  4>(jc,  _y,  2)  is  a  solution  of  the 
complete  system  Avf=  o  and  ^./—  o,  the  equations 


give       :     :    =' 


Since  the  total   differential  equation  which   has  c/>(.\,  r 
for  solution  is  . 


114  THEORY   OF   DIFFERENTIAL   EQUATIONS  §33 

or  differs  from  it  by  a  factor  involving  ,r,  y,  z  only,  this  equation 
may  take  the  form 

(87)  «?A-  Q^dx+(R&-Rfddy+(P^-PiQdti=** 

The  problem  of  solving  a  complete  system  is  thus  reduced  to  that 
of  solving  a  total  differential  equation  (87).  At  times  the  actual 
work  involved  in  solving  (87)  turns  out  to  be  simpler  than  that  re 
quired  by  the  method  of  the  previous  section. 

Besides  the  usual  methods  for  integrating  total  differential  equations  (see 
F.L  Dif.  Eq.  Chapter  VI)  the  following  method  due  to  Dubois-Reymond  may  be 
mentioned. 

Instead  of  letting  one  of  the  variables,  say  c,  be  a  constant  temporarily,  as  is 
usually  done,  let  it  be  a  linear  function  of  the  other  two,  thus 

z  —  x-^ay 

where  a  is  an  arbitrary  constant.     This  relation  carries  with  it 

dz  —  dx  -\-  ady. 

Eliminating  z  and  dz  from  these  two  and  the  total  differential  equation,  there  re 
sults  an  ordinary  differential  equation 

Jlf(xt  >',  a)itx  +  A\x,  y,  a}dy  =  o  f 
whose  solution  \}/(x,  y,  a}  =  const. 

*  Equation  (87)  may  be  put  in  the  convenient  determinant  form 
dx    dy     dz 
(?i     A'j 
Q-i    RI 
which  expresses  the  condition  that  the  above  three  homogeneous  linear  equations  in 

^,    2*.    ^  are  consistent. 
dx     dy      dz 

t  If  it  happens  th.it  this  differential  equation  does  not  contain  a,  some  other  linear 
relation  among  the  three  variables  containing  an  arbitrary  constant  should  be  tried 
leading  to  a  differential  equation  in  two  of  the  variables  only  and  containing  the  arbi 
trary  constant. 


§§33,34      LINF.Ak    PARTIAL  DIFFERENTIAL    KOUATIONS 
gives,  on  replacing  a  by  its  value  in  terms  of  x,  y,  z, 

=  const., 


which  is  the  solution  of  the  total  differential  equation. 

This  method  requires  the  solution  of  only  one  ordinary  differential  equation 
instead  of  two,  as  in  the  usual  method,  when  an  integrating  factor  is  not  known. 
But  in  actual  practice,  this  theoretically  simpler  method  may  not  prove  as  de 
sirable  as  the  other. 

Ex.  The  examples  of  §  32  should  be  solved  by  the  methods  of 
this  section. 

Thus  for  Kx.  I  the  total  differential  equation  to  be  solved  is 


dx     d      dz 


o       o 


x      y 
Its  solution  is  -^  =  const. 


=  z  dy  -  ydz  =  o. 


For  Ex.  2 


dx  dy  dz 

x  y  z 

x-  +  y1  +  yz     x2  -f  y2  —  xz     z(.r  +  y 


=  0 


becomes,  on  multiplying  the  second  row  by  x  +  y  and  subtracting  from  the  third 

row, 

dx      dy      dz 

x       y        c    =0 

.  y    —x    o 

xz  dx  +  yz  dy  —  (x~  -f  y2)dz  =  O. 


An  obvious  integrating  factor  is 


*(•** 


,  and  the  solution  is  --      ^  =  const. 


34.    Linear  Partial  Differential  Equation  Invariant  under  a  Group.  — 
The  homogeneous  linear  partial  differential  equation  of  the  first  order, 


(88) 


Il6  THEORY   OF   DIFFERENTIAL   EQUATIONS  §34 

has  two  independent  solutions  <l>i(x,  y,  z)  and  <f>o(x,  y,  z).  Every 
other  solution  is  some  function  of  these. 

The  result  of  transforming  (88)  by  the  transformation 

(  89)  xj_  —  <f>  (x,  y,  z),  }\  =  \j/(x,  y,  z),   zv  =  ^(x,  y,  z) 

is  ([15],  §  n)  the  new  equation 

(90)  ^+^+^1=0; 

wliere  A<j>,  A\l/,  A%  are  to  be  expressed  in  terms  of  xlt  ylf  %.  If  (90) 
is  the  same  equation  in  the  new  variables  as  (88)  is  in  the  old  ones, 
or  differs  from  it  by  a  factor,  the  transformation  (89)  is  said  to  leave 
the  differential  equation  (88)  unaltered.  In  this  case  it  must  trans 
form  both  <#>!  and  <£2  into  solutions  again  ;  that  is,  they  are  either  left 
unaltered  by  (89)  or  they  are  transformed  into  some  functions  of 
themselves  by  it. 

Let  us  find  under  what  condition  (88)  is  left  unaltered  by  every 
transformation  of  the  group 


We  have  seen  ([7],  §  n), 

«kC*i,  v,,  *,)  =  *,.(.v,  y,  z)  +  U^  '-  +  LT^  £  -f  .- 


In  order  that  this  be  a  function  of  ^Lv,  v.  :  )  and  <^L,('.v,  r,  z)  for  all 
transformations  of  the  group,  i.e.  for  all  values  of  /,  it  is  necessary  that 

tf<fc=/X<h,   «fe)f  (1=1,   2). 

It  is  readil    seen  that  this  is  also  a  sufficient  condition.      For 

67W,,  ^  =         Ufr  -h         u^  =  WF}  +       .  p^ 


LINEAR   PARTIAL   DIFFERENTIAL   EQUATIONS 


117 


which  is  again  a  function  of  fa  and  fa.  In  the  same  way  it  can  be 
shown  that  if  (Jkcf>i  is  a  function  of  fa  and  fa,  (7k+l<f>i  is.  Hence, 
(91)  is  the  necessary  and  sufficient  condition  that  the  equation* 
whose  solutions  are  fa  and  fa  shall  be  invariant  under  the  group. 

It  is  desirable  to  have  a  condition  expressed  in  terms  of  the  differ 
ential  equation  itself.     The  linear  equation 

(92) 


has  fa  and  fa  for  solutions  when  A/=  o  is  invariant  under  Uf.     For 
(UA)fa  =  L/Afa  -  AC/fa  =  U(o)  -  AF^fa,  fa)  =  o 
(1=1,2). 

Since  (88)  and  (92)  have  the  same  solutions,  they  must  be  the 
same  equation,  to  within  a  possible  factor,  by  the  previous  footnote. 

*  A  unique  linear  differential  equation  of  the  form  (88)  (to  within  a  possible  factor 
involving  the  variables  only)  is  determined  by  two  independent  solutions.  For  if 
0!  and  <J>.2  are  the  solutions  of 

4fte/»£+.0if  +*£-* 

dx        dy         -dz 


then 


whence 


.  p  _ 


d.y 


So  that  the  differential  equation  having 
convenient  form 


=     -=  --  r       .--=—  =    , 
dx          By  62 

-  0^0-2  i  ^  ^0-2  i  r>  dfa 
=P-=  —  H  Q  -^  -  +  R  -7;  —  =  o, 
dx          dy  dz 

'  f^2  •  ^1  dfa  __  501  d<t>-i  .  50i  50-_>  _  (901  502 
s    dy      d*    dx       dx  dz   '  3-»-    (>       3r   ^-^ 

and  <.>  for  solutions  ma    be  written  in  the 


dx  dy  d* 

50i  301  a^ 

3^  3;- 

502  50-2  30- 

dxt  dy  d= 


Il8  THEORY   OF   DIFFERENTIAL   EQUATIONS  §34 

Hence  when  Af—  o  is  invariant  under  Uf 
[31]  (UA)f=\(X,y,z)Af. 

Conversely,  when  [31]  holds,  (88)  and  (92)*  have  the  same  solu- 
tions;  then  =  -o,  (1  =  1,  2). 


Since  A<^i  =  o)  it  follows  that  A  (Ufa)  —  o  ;  hence  C/fa  is  a  solution  of 
(88),  and  must  be  a  function  of  fa  and  fa. 

Therefore  [31]  is  both  the  necessary  and  sufficient  condition  that 
Uf  leave  Af  =  o  unaltered.^ 

Thus,  the  group  Uf=x  &  +y  |f  +  z  &  leaves  J/&&+&+  |/=  o  unal 

tered,  since       (U^)f~-  (§1  +  |/  +  |A  =_  ^ 
Vd*     6>>'     6s  / 

Similarly  the  same  group  leaves  Af=y  J-  —  x~  +  z^=o  unaltered,  since 


It  also  follows  from  this  that  the  group   Uf=y  |C  -  *  ?/  +  2  Cleaves   the 

5/"         5/         3/  ^         ^^ 

equation  ^4/=  x  -^-  +  y  ~  -\-  zjt  =  o  unaltered. 

Remark.  —  From  the  form  of  the  condition  [31]  it  is  obvious  that 
if  an  equation  Af  =  o  is  invariant  under  each  of  a  number  of  groups 
£/,/,  £/>/,  •••,  Urft  it  is  invariant  under  Uf=alUlf+a,U,f+  ••• 
-f  arUrf,  where  alt  a.,,  •••,  ar  are  any  constants. 

*  If  \(x,  y,  z)  is  identically  zero,  in  other  words  if  (L?A)/=o  for  all  functions  f, 
[31]  is  still  a  sufficient  condition  that  Uf  leave  Af—o  unaltered.  In  this  case  one 
cannot  speak  of  the  equation  (92);  but  writing  the  identity  ((J.-f)/~~o  in  the  form 
UAf^AUf,  it  follows  that  AU^  =  Q  since  UAQi  :  =  U(o)  =  o.  Hence  U^  is  a 
function  0i  and  <t>->  as  above. 

t  Using  the  method  of  the  previous  footnote,  it  can  be  shown  that  a  homogeneous 
limMr  equation  in  n  variables  is  determined,  to  within  a  factor  by  its  n  —  i  independent 
solutions.  The  argument  of  this  section  therefore  applies  without  change  to  such  an 
equation.  Hence  [31]  is  the  condition  that  Af—  o,  involving  n  i\viat<les,  shall  be 

invariant  under  the  group  Uf     £i   /'      I   &  „      +  '"    '  ^.  ^  '      In  ?  X5  essentially  the 

Qx\          (j/.vo  ('!»,, 

same  method  for  the  case  of  two  variables  was  carried  out. 


§§34,35      LINEAR    PARTIAL    DIFFERENTIAL   EQUATIONS  I  19 

Ex.     Determine  which  of  the  equations  below  are  left  unaltered 
by  each  of  the  following  groups  : 


dx 


2.  iy=-* 

ox         dy 


3.    V/=* 

dx 


4. 


<1f        .,  df  .  df 

d.Af=vfx-rfy+(*+flj=<>. 

35.    Method  of  Solution  of  Linear  Partial  Differential  Equation  In 
variant  under  a  Group.  —  If  the  equation  Af=  o  is  invariant  under 

Vf 

[31]  (UA\f=\Af, 

i.(\  IJ—  o  and  Af=o  form  a  complete  system.     Hence  the  methods 
of  §§  32  and  33  are  available  for  finding  one  of  the  solutions  of  Af—  o. 

*  While  lr/=P(*ty,  c).  //"leaves  ///"—  o  unaltered  for  all  forms  of  p(r,y,  c),  such 
a  group  is  said  to  lie  //•/:•/,//  because  it  is  «»t  no  service  in  solving  Af—  o.     We  shall 

presuppose  that  tin-  group  //under  consideration  here  is  not  trivial. 


120  THEORY   OF  DIFFERENT]  AL   EQUATIONS  §35 

Having  thus  found  $(x,  y,  z),  a  common  solution  of  6^=0  and 
Af=o,  a  second  solution  of  A/=o  may  be  found  in  the  followin'g  way  : 

Since  <(>(x,  y,  z)  is  not  a  constant,  it  must  involve  at  least  one  of 
the  variables,  say  z.  Replacing  z  by  the  new  variable 

z=<l>(x,}',  z), 
the  equation  and  the  group  take  the  forms  ([15],  §  n), 

Af=  P(x}  y,  z}^  +  Q(x,  y,  *')|^=  o, 
ox  dy 


Uf  =  |(>,  yt  z) 

since  A$  =  o  and  U$  =  o. 

Here  P,  Q,  |,  T\  are  what  P,  Q,  £,  -q  respectively  become  when  in 
them  z  is  replaced  by  its  value  in  terms  of  x,  y,  z  obtained  from 
z  =  <j>(x,y,  z).  Here  z  plays  the  role  of  a  constant  since  the  coeffi 

cients  of  —  in  A  /and  Uf  are  both  zero.     To  solve  Af=o  we  pro- 
dz 

ceed  to  the  corresponding  ordinary  differential  equation 
Qdx  —  Pdy  =  o. 

This  is  invariant  under  Uf.     Hence  the  methods  of  §§    12  and  20 
may  be  employed. 

Remark.  —  When  the  usual  Lagrange  method  (see  RL  Dif.  Eq. 
•§  79)  is  practicable,  it  will,  as  a  rule,  prove  simpler  than  the  method 
of  this  section.  As  an  exercise  it  may  be  desirable  to  solve  the 
examples  below  by  both  methods.  But  the  Lie  method  is  of  inter 
est  theoretically  and  may  prove  valuable  when  the  other  method  can 
not  be  carried  out. 


Ex.  1.    Af~  2  xv-       -  2  *y+(/-  x'*)z=  o. 
dx  '    (>v  dz 

The  coefficients  are  homogeneous  and  of  the  same  degree.     Hence 

this  equation  is  left  unaltered  by  the  group  Uf  '=  x  -;'-  -f  v  -£-  +  *-£•  ', 

i).\-     ~  dy        dz 
as  a  matter  of  fact,  (L7A)f  =  2  Af. 


§§35.  36      LINEAR   PARTIAL  DIFFERENTIAL  EQUATIONS  121 

xv 
By  the  method  of  §  32  or  that  of  §  33,  -^  is  readily  found  to  be 

z~ 

the  common  solution  of  Af  =  o  and  Uf=  o. 

The  transformation  z  =  -+-  reduces  A/=o  and  Uf  to 


Af  =  2  xy- -J-—  2 x2y  ^~  =  o  and  Uf  =  x-^-+y-^- 
dx  dy  dx     '   dy 

respectively.     The   corresponding   ordinary    differential   equation  is 
2  x2y  dx  H-  2  xyz  dy  =  o. 

Lie's   integrating    factor    -?- (or  the  obvious  integrating 

2xy(x*+y*)    \ 

factor  --  ]  leads  at  once  to  the  solution  x2  -f-j^  =  const.     Hence  two 


independent  solutions  of  Af=  o  are  5|  and 

z~ 

dx  /  dy  '  '  dz 


Ex.3.    ^=^  +  v 

oy 

Invariant  under    Uf  =  (x  +}')--  +(^  +  v)—  -f  2  z^-,  as  well   as 

dy  dz 


*y\ 

Ex.4.    Af  =  (xz-ti&    . 

dy 

[Invariant  under  Uf=x+-  +  v—  • 
dx        dy_\ 

36.  Jacobi's  Identity.  —  For  further  development  of  the  theory  it 
will  be  necessary  to  have  available  a  certain  identity  first  noted  by 
Jacobi  and  known  by  his  name  : 

If  Aif,  A^fj  A^f  arc  three  homogeneous  linear  partial  differential 
expressions  in  any  number  oj 

(93) 


122  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§36,37 

This  may  be  verified  directly  in  the  case  of  three  special  forms, 
and  also  in  the  general  case  for  two  variables.  This  is  suggested  as 
an  exercise  to  the  student. 

Probably  the  simplest  way  to  prove  the  theorem  is  the  following, 
due  to  Engel  : 

Since  (A^A^f  ==  A^f-  A,AJ, 

((A,A,)A,}f~  A,A2A,f-  A.A.A.f-  A.A^f  +  A^A.f, 

A.A.A,/, 


The  sum  of  these  is  obviously  identically  zero.     Hence  the  identity 
(93)  is  established. 

37.  Linear  Partial  Differential  Equation  Invariant  under  Two 
Groups.  —  If  the  equation  Af=  o  is  invariant  under  two  distinct* 
groups  UYf  and 

[31'] 


Jacobi's  identity  (93)  for  UJ,  U^f,  A/  is 


Using    [31']    and    obvious    properties   of    alternants    (§  14),    this 
bec°mes 


where  /x  =  Uv\^  —  U<X\.     Hence  the 

THEOREM.  —  If  Af  =  o  is  invariant  under  U\f  and  £/,/,  //  is  also 
invariant  under  (£/,  6o)/.f 

*  Two  groups  £/!/  and   U^f  are  said  to  be  distinct  with  respect  to  the  equation 
/(/—  o,  provided  no  relation  of  the  form 

(94) 


exists,  where  al  and  a.2  an-  constants  and  p  is  any  function  of  the  variables.  For  it  is 
obvious  that  if  (  \  f  leaves  .  //  o  unaltered,  U.,fi-icU\f-\-  pAf  will  also  do  so  for  all 
choices  of  the  constant  c  and  of  the  function  p(x,y,  z). 

f  This  theorem  holds,  and  is  proved  in  exactly  the  same  way,  for  n  variables. 


37 


LINEAR    PARTIAL   DIFFERENTIAL   EQUATIONS 


123 


If  (£/;£/,)/  is  not  of  the  form 
(95)  *iUJ  +  aMf  +  P(x,  y,  z)Af, 

where  a±  and  a2  are  any  constants  and  p  is  any  function  of  the  vari 
ables,  it  is  said  to  be  distinct  from  U\f  and  U^f  with  respect  to  the 
equation  Af  =  o.  In  this  case  the  theorem  gives  a  new  group  under 
which  the  equation  is  invariant.  The  theorem  may  then  be  applied 
to  this  new  group  and  one  of  the  original  ones.  And  so  on. 

Remark.  -—  It  is  important  to  note  that  there  always  exists  a  linear 
relation  between  four  homogeneous  linear  partial  differential  expres 

sions  of  the  first  order  in  three  variables.*     For  eliminating  -£.      *- 

dx     dy 

-•-  from  the  four  identities 


* 

=  ft  -7- 

ox 


-  -         :    , 

dy         dz 


' 

dz 


the  linear  relation 


U,f    ^     r,, 


ft      17, 


is  obtained.     In  general  the  coefficients  are  functions  of  the  variables. 


*  Similarly,  there  is  always  ;i  linear  n^laticn  between  n  -\- 1  sueli  expressions  in  n 
variables. 


124  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§37,38 

As  a  consequence  we  always  have 


If  it  turns  out  that  «j  and  «2  are  constants,  this  is  of  the  form  (95),  in 
which  case  (£/i&Q/is  not  distinct  from  £^/and 


Thus  the  equation  Af&&  +  &  +  &=  o  is  left  unaltered  by 
d*      dy      dz 


dy  dx         dy 

since  (£/i/l)/=o,    (U*A}f=-  2  x  Af. 


Moreover      (U^U^f=-(x  -  y~]        ^  2(y  -  z)(x  - 

dy  z 

also  leaves  Af=  o  unaltered,  since  ((U\U*)A)f=o.     It  is  readily  seen  that 


Again 

'  4-  2(7  -«)(*-  2 


x—y      \x-y 


also  leaves  Af  —  o  unaltered,  as  is  readily  verified.     And  so  on. 

38.  Methods  of  Solution  of  Linear  Partial  Differential  Equation 
Invariant  under  Two  Distinct  Groups.  —  Two  important  cases  are  to 
be  distinguished  : 

A*.    If  a  relation  of  the  form 

(9 7)  UJ=  u(x,  y,  z)  UJ-  +  p(x,  y,  z)Af 


§38  LINEAR   PARTIAL   DIFFERENTIAL   EQUATIONS  125 

exists,*  where  a  is  not  a  constant,  6^/is  still  considered  distinct  from 
Uif.  In  this  case  a(x,  y,  z)  is  a  solution  of  Af=  o.  For,  since 
Af=  o  is  invariant  under  U^f,  (£72^)/must  be  a  multiple  of  Af.  But 


-  AP  Af 
=  (a\l-AP)A/-Aa  UJ. 

Since  U\f  is  supposed  to  be  not  trivial,  i.e.  not  a  multiple  of  Af 
(§  35  )>  the  onty  waY  in  wnicri  (£4<4)./can  be  a  multiple  of  Af  is  by 
having  Aa  =  o.  Hence  «  is  one  of  the  two  independent  solutions 
of  Af=  o  to  be  found,  f 

To  find  a  second  solution  of  Af=  o,  several  possibilities  may  arise 
which  will  be  mentioned  in  the  order  of  desirability  : 

i°  Since  A/=o  is  invariant  under  U^f,  67i«  is  also  a  solution  of 
Af=o  [(91),  §  34].  If  U\<t  turns  out  to  be  distinct  from  a,  it  may 
be  taken  as  the  second  solution  necessary  to  give  the  general  solu 
tion  of  Af—  o. 

2°  If  U\u,\  is  a  function  of  a  or  a  constant  other  than  zero,  two 
methods  are  possible  : 

*  A  linear  relation  between  Af,  U±f,  C/2/will  show  itself  by  the  vanishing  of  the 
determinant  of  their  coefficients,  thus 


A  = 


P       Q       R 

~io. 

»?2 


Here 


t  Conversely,  if  a  is  a  solution  of  A/=o  and  f/i/is  a  group  that  leaves  the  equation 
unaltered, 
[35]  U 


will  also  leave  it  unaltered  no  matter  what  be  the  form  of  p(x,  y,  z).     For 

(  U2A)/=  (a  f/i  +  pA,  A)/=  (a\i  -  Ap)Af. 
since  Aa=o.     [Compare  (35)  §  17.] 

I  Since   (S.2  a  =  a  U±  a  -f-  pAa.  =  a  U\.  a,  it  is  sufficient  to  consider  U{a  only. 


126  THEORY   OF  DIFFERENTIAL  EQUATIONS  §38 

(a)  The  solution  common  to^//=:o  and  £/i/~=o  (or  £72./=  o) 
may  be  found  by  either  the  method  of  §  32  or  that  of  §  33.  Since 
£/,«  ^=  o,  this  common  solution  will  be  independent  of  a. 

(fr)  Since  a  must  contain  at  least  one  of  the  variables,  say  z,  the 
introduction  of  the  new  variable  z  =  a  (x,  y,  z)  in  place  of  z  reduces 
A/  =  o  to  one  in  two  variables, 


z  appearing  as  a  constant  since  the  coefficient  of  -*-  is  zero.     (Com- 

dz 

pare  §  35.)     But  since    L\u=f=  o,  the  above  equation  must  be  inte 
grated,  without  any  further  assistance  from  the  groups  U\f  and  U.2f. 
3°.    If  £/i«  =  o,  the  method  of  §   35  is  available  ;  thus  the  intro 
duction  of  the  new  variable  z  gives  the  same  differential  equation  as 
above,  but  now  the  transformed  group 


under  which  it  is  invariant  also  leaves  z  unaltered.  Hence  the 
methods  of  §§  12  and  20  are  available  for  solving  the  corresponding 
ordinary  differential  equation 


B.    If  no  relation  of  the  type  (97)  exists  between  Aft  U±ft  U>>f, 
the  relation 

(96)     (UiU*)f=  a,(x,y,  z)l\f+  «,(*,;•,  *)  *V+  P  (*,;•,  z}Af, 

which  always  exists  (Remark  §  37),  will  prove  of  service  if  «t  and  a.2 
are  not  both  constants  ;  for  a^  and  a.,  are  solutions  of  Af—  o,  as  may 
be  seen  from  the  following  consideration  :  * 

*  By  exactly  the  same  kind  of  'reasoning  as  that  employed  here,  the  following  gen 
eral  theorem  can  be  established.     (It  is  suggested  that  the  student  carry  out  the  proof.) 
Jfthe  equation  in  n  -variables 


§38  LINEAR    PARTIAL   DIWKKKNTIAL    EQUATIONS  I2/ 

By  the  Theorem  of  §  37,  (  £/i  £/,)/  leaves  Af=  o  unaltered.     Hence 


But 

a,(U,A)f- 


u,\.,  —  Ap)A/—  Auv  L\f 

Since  no  linear  relation  is  supposed  to  exist  between  Af,  U\f,  U>>f, 
the  only  way  in  which  ((C/tCfyA)/  can  be  a  multiple  of  Af  is  by 
having  Aa^  =  o  and  Aa.2  —  o.  Hence  rq  and  a.2  are  solutions  of 
Af=  o.* 

i°  If  «,  and  a.,  are  two  independent  functions  of  the  variables,  the 
general  solution  of  A/=o  is  known  witliout  any  further  work. 

2°  If  one  of  them,  say  a  ,  is  a  function  of  the  variables,  while  the 
other,  a.,,  is  either  a  function  of  av  or  a  constant,  use  may  be  made  of 
the  fact  that  £/i«i  and  6£«j  are  also  solutions  of  A/=o  [(91),  §  34]. 
If  either  of  these  turns  out  to  be  a  function  distinct  from  alt  it  i«ay 
be  used  as  the  second  solution. 


/j  invariant  under  r  -f-  I  distinct  groups  U^f,  U.±f,  •••,  Ur+\f,  and  if  no  linear  relation 


exists  between  .If  and  r  of  the  lr/'st  but 

Ur+l/= 
then  MI,  Ct-i,  •••,  ctr  are  solutions  of  Af=  o. 

*  The  student  should  have  no  difficulty  in  showing  that,  conversely,  if  «,  and  «o 
are  solutions  of  Af=  o,  and  6^/"and  £/>/"are  two  groups  that  leave  the  equation  unal 
tered,  the  group 
[35'] 


will  also  leave  it  unaltered  no  matter  what  be  the  form  of  p(v,y,  z).     (Compare  [35] 
above.) 


128  THEORY   OF   DIFFERENTIAL   EQUATIONS  §38 


3°  If  both    £/!«!   and    £/>«i*  are   either   functions   of  nv  or  con 
stants,  either  of  the  methods  (a)  and  (If)  of  A,  2°  may  be  employed. 
Or, 

(a)  if  one  of   U&\  and    £72«i  is   zero,  the   method  of  A,  3°  is 
available, 

(b)  if  neither  is  zero,  the  group    Vf=  U^U^f—  U^U^f  leaves 
Af=  o  unaltered,  and  Va.^  =  o  ;  hence  case  (a)  exists. 

4°  If  both  «t  and  «2  are  constants,  say  a^  and  a2,  the  solution  com 
mon  to  Af=  o  and  U\f=-  o,  and  that  common  to  Af—  o  and 
[/2f=o  may  be  found  by  either  of  the  methods  of  §§  32  and  33. 
Moreover,  these  solutions  will  be  independent  since  there  is  no 
linear  relation  connecting  Af,  L/\ft  £/>/.  (Theorem  II,  §  31.)  We 
shall  show,  by  a  method  due  to  Lie,  that  an  integrating  factor  for  at 
least  one,  and  sometimes  for  tfoth,  of  the  total  differential  equations 
arising  in  the  method  of  §  33  can  be  found  in  this  case.  (But  it  is 
possible,  at  times,  to  find  by  inspection,  an  integrating  factor  that  is 
simpler  than  the  one  given  by  the  following  method)  : 

In  (  Ui  U.2  )/=&i  U\f+a2  Uzf  +  p  A/  either  al  and  a2  are  both  zero 
or  they  are  not. 

(a)  If  *!  =  a,  =  o,          (  U,  Uz}f=  PAf. 

Since  Af=  o  is  invariant  under  [/.2f, 


If  <£(je,  j,  z)  is  the  common  solution  of  Af=  o  and  6r1/=  o, 

=  o,  since  (  U\  (72)<f>  =  U\  U<&  —  U^  U^  =  pA<}>  =  o  ; 
=  o,  since  (  U«A)$  =  U»A<$>  —  A  U<&  =  \2A<f>  =  o. 


These  identities  can  hold  only  provided    £/2<£  is  a  solution  of  both 
—Q  and  Af=o;  i.e.    U.&  must  be  a  function  of  <f>,  say 


*  In  this  case  (  U^U^d.^  will  also  be  a  function  of  U^  or  a  constant,  including  zero, 
because  of  (96). 


LINEAR    PARTIAL   DIFFERENTIAL   EQUATIONS 


129 


Moreover  jF(<£)  ^  o,  for,  as  noted  above,  Af=  o,  Uvf=  o,  U»f=-  o 
cannot  have  a  common  solution,  since  they  are  independent.  As 
was  done  in  an  analogous  case  in  §  12,  </>,  the  common  solution  of 
Af~  o  and  L\/=  o,  may  be  chosen  in  such  a  form  that  £/><£  =  i  .  It 
must  then  satisfy  the  three  equations 


dx 


dy 


dy 


a-r         r)2- 
dx  dy 

-* 


dz 


These  equations  determine  -S  -,  -     }  whence  <^>  is  obtained  from 


dx     dy     dz 

<l<l>=*±dX  +  *+<ly+d-+t 
dx  dy  dz 


by  the  quadrature 


\dx  dy  dz 
P  Q  R 
6  171  Ci 


,  where  A  = 


£>          !/•>        £2 


In  exactly  the  same  way,  ^,  that  form  of  the  common  solution  of 
Af—  o  and  L7.if=o  for  which  C^  =  —  i,  may  be  obtained  by  the 
quadrature 


The  determinant  A  is  thus  seen  to  be  an  integrating  factor  for  each 
of  the  total  differential  equations  arising  in  the  method  of  §  33  for 
finding  the  two  independent  solutions  of  Af=  o. 


130  THEORY   OF   DIFFERENTIAL   EQUATIONS  §38 

(<£)  If  only  one  of  a^  and  a.,  is  zero,  let  a2  =  o.     Then 


In  precisely  the  same  way  as  before,  tt  is  seen  that,  if  <£  is  the 
common  solution  of  Af  =  o  and  U-^f  =  o,  U<$>  =  -/yX0)  ^  °-  Hence 
that  form  of  <£  for  which  U«$  ~  i  is  given  by  the  (quadrature 

dx      dy      dz 

r     Q    R 

&         rj}        & 


To  find  a  second  solution  of  A/  =  o,  independent  of  <j>,  either  the 
method  of  A,  3°  may  be  employed,  or  the  common  solution  of 
Af  =  o  and  £/2/=  o  may  be  found  by  one  of  the  methods  of  §§32 
and  33. 

(f)  If  both  (7i  and  a2  are  different  from  zero,  consider  the  two 
groups 


These  are  obviously  distinct  and  leave  Af  =  o  unaltered.     More- 


We  are  thus  under  case  (/;)  and  the  method  for  that  case  may  be 
employed. 

Note.  —  For  practical  purposes  it  may  be  worth  noting,  that  the 
choice  of  the  groups  U^f  and  Vf  =  a^  U^f  +  ti.2  U^f  also  leads  to 
case  (f). 

Remark  i.  —  A  hasty  survey  of  the  processes  involved  in  the 
methods  to  be  employed  in  the  various  cases  considered  in  this  sec 
tion,  brings  out  the  fact  that  when  two  distinct  groups  are  known 
under  which  the  equation  Af  =  o  is  invariant,  the  solution  of  the 
latter  can  be  obtained  by  means  of  quadratures  only,  except  in  the  case 
of  A,  2°,  where  one  ordinary  differential  equation  of  the  first  order 


§38  LINEAR    PARTIAL   DIFFERENTIAL   EQUATIONS  131 

must  be  solved.  In  certain  cases,  such  as  A,  i°  and  B,  i°  and  2°,  no 
integration  whatever  is  required.  In  the  above  scheme,  certain  alter 
native  methods  involving  the  solution  of  differential  equations  have 
also  been  suggested,  for  in  certain  cases  these  processes  may  prove 
simpler  of  execution  than  those  involved  in  carrying  out  quadratures. 
Remark  2.  —  It  is  easy  to  prove  the  existence  of  a  pair  of  groups 
U^f  and  U^f  under  which  A/=o  is  invariant,  and  for  which  no 
linear  relation  of  the  form  (97)  holds.  For  Af  =  o  has  two  indepen 
dent  solutions  (j)l  and  <£2.  These  are  independent  with  respect  to 
at  least  two  of  the  variables,  say  x  and  y.  Introducing  the  new 

variables  .  / 

x  =  $!(#,  yy  «),    y  =  £(«,  j,  2),    z  =  z, 

Af  =  o  takes  the  form 


=  0. 

dz 

By  inspection  C/i/=  -*-  and  U.,f  =  -^-  are  seen  to  leave  the  differ 
ed  dy 

ential   equation  unaltered.     Moreover  there   is  obviously   no   linear 

relation  between  -J-,  SL,  §£•     Passing  back   to  the  original  varia- 
dx    dy    dz 

bles,  Af=o  will  be  invariant  under  the  groups  (/if  and  U»f  into  which 
U}/  and  Z7,/  are  transformed,  and  no  linear  relation  can  exist  now. 

Ex.  1.     ^/  =  (.v  +  v.,     v"-.r  2S/= 

This  equation  is  invariant  under 


U,f  =  ,(.v  +  ,,f       +,(*  +  j02+  [(.vv  +  2  ^)(.v  +  v)  +  4  .vvs] 
as  may  be  verified  easily. 


132  THEORY   OF   DIFFERENTIAL   EQUATIONS  §38 

Here  A  =  o,  and  U*f  =  (yz  +  zx  +  xy)  UJ  -  xyAf,  (A). 

.\yz-\-  zx  -f  xy  is  a  solution  of  Af  =  o. 

Moreover  U^yz  +  zx  +  xy)  =  $(yz  +  zx  +  xy)  +  (x—  y)2  is  also  a 
solution  (A,  i°). 

Taking  account  of  the  first  solution,  the  second  one  may  be 
replaced  by  x  —  y.  Hence  the  general  solution  of  Af=  o  is 

<&(yz  +  zx  -h  xyt  x  —  y)  =  o. 

Ex.  2.    Af^xz-y^  +  ^z-x^  +  ^-z^-^v. 
ox  oy  oz 

This  equation  is  invariant  under 

9f     '  df 
=x-£-+yJ-  and 

dx         dy 


A  =  o,  and  £/>/  =  (, 
.*.  ^  -f-  yz  is  a  solution  of  Af  •=•  o. 
Moreover  ^(jc  +yz)  —  x  +yz  ^  o  (A,  2°). 

To  find  the  solution  common  to  Af=o  and  [\/=o  the  method 
of  §  33  requires  the  solution  of  the  total  differential  equation 

y(i  —  z~)dx  —  x(i—  v)dy  +(yi  —  x*)dz  =  o. 

An  obvious  integrating  factor  is   -  -  -  ,  leading  to  the 

(f  —  * 

solution 


- 
\x-yi  ~z 


x  -\-yz-  (y-\-xz) 
The  left-hand   member  of  this  is,   therefore,  a  second  solution 


§38  LINEAR   PARTIAL   DIFFERENTIAL   EQUATIONS  133 

Taking  account   of  the    first    solution,   the    second   one   may  be 
replaced  by  y  +  zx.     Hence  the  general  solution  of  Af=o  is 


$>(x  +yz,  y  +  zx)  =  o. 
Ex   3.    A=x 


ox  oy 


OZ 

Ex.  4.    A= 


dx         dy          dz 

E,5.   Af=f  +  !f  +  ¥  =  0. 
ox      o         oz 


EX.   6.      Af=  (xz  -  y)        +  (yz  -  X)       +  (  I  -  Z*)       =  O. 
dx  dy  dz 


CHAPTER   VI 

ORDINARY    DIFFERENTIAL    EQUATIONS    OF    THE    SECOND 

ORDER 

39.    Differential  Equation  of  the  Second  Order  Invariant  under  a 
Group.  —  The  differential  equation  of  the  second  order 

(98)  >'"=f(x,y,y') 

is  equivalent  to  the  system  of  equations  of  the  first  order  * 

(99)  *k  =  </v  =  __  *  __ 

i      /     S(*,y,y') 

If  the  solutions  of  the  latter  are 
(  i  °o)  //  (.v,  r,  r  ')  =  «,   ?>O,  j-,  ]•')  =  /*, 

the  solution  of  (98)  may  be  obtained  by  eliminating  y'  from  the  two 
equations  (loo).f 

Instead  of  solving  (99),  one  may  find  //  and  v  as  two  independent 
solutions  of  the  corresponding  linear  partial  differential  equation  J 


The  problem  of  solving  (98)  is  thus  reduced  to  that  of  finding  two 
independent  solutions  of  (101). 

If  (98)  is  invariant  under  a  group  Uf,  the  equivalent  system  (99), 
involving  the  three  variables  x,yt)\  is  invariant  under  the  extended 

*  El.  /)>/,  /-.-/.  $  68. 

f  Tin-    c(iuations    (too)   ;in:    two    independent    first   integrals  of    (98).      (Sec  f  52, 
Theorem  IV.)  t  M-  !>'/•  l'-'l-  k  79- 

134 


§39  ORDINARY,   OK  THE   SECOND   ORDER  135 

group  U'f.  The  effect  of  U'f  on  ?/  and  r  is,  therefore,  to  trans 
form  them  into  some  functions  of  themselves  ;  i.e.  £/'//  —  <£(//,  v\ 
f/'v  =  {f/(u,  v).  Hence  the  linear  partial  differential  equation  (101) 
having  u  and  v  for  solutions  is  also  invariant  under  U'f(§  34). 
'Consequently  the  method  of  §  35  may  be  employed  to  find  u  and  v. 

Remark.  —  Since  the  invariance  of  (98)  under  Uf  implies  the 
invariance  of  (101)  under  the  extended  group,  and  conversely,  it 
follows  from  the  remark  of  §  34  that  if  (98)  is  invariant  under  each 
of  a  number  of  groups  U^f,  U<>f,  •••,  Urft  it  is  invariant,  under  the 
group  Uf=  alUlf+  a.1U~.1f+  •  ••  -\-  arUrf,  where  «,,  a.,,  •••,  ar  are 
any  constants. 

This  remark  applies  without  modification  to  a  differential  equation 
of  any  order,  because  the  form  of  the  condition  [31],  §  34  is  inde 
pendent  of  the  number  of  variables  appearing  in  the  linear  partial 
differential  equation  Af=  o. 

EX.  i.  x 


This  equation  is  invariant  under  Uf=  x  -J-  +  ny  -*-  for  any  value 

dx          dy 

of//  (VI,  §  28).     In  particular  it  is  left  unaltered  by  Uf^x^-^y^-. 

d.v         dy 

Here  Af^  %  +y>  gf+^-  V)  If  =  0. 

dx          dy  xy         dy' 


dx     *  dy 
For  the  method    of  §  32*  use   may  be   made  of  the    fact   that 

u  =  ^,  v  =  v'  are  solutions  of  U'f=  o. 
.v 

*  The  method  of  $  33  requires  the  solution  of 
i/r     </Y  rtV' 

X  r— o,   or   (_y  —  .n'')( —       l  4«*   r  +</v'j  =  o. 

^        .v  y  I 

y  o 

The  evirlent  integrating  factor  leads  to  the  solution  -vv        avfj/. 

/(>"•*•/)  x 


136  THEORY   OF  DIFFERENTIAL    EQUATIONS  §39 

Au—xy'—y      .^..-/P'  —  */)         .  Av  _       XV9  V 

All—      -  -  -  ,     AV  —  —  —  .       .   .  —  —  —  --  •-—  =  --- 

x-  xy  An  y  u 

Equation  (86),  §  32  is         ^_^£^=0>     ...  p=  uv  =  JZ'. 
du       u  dv  x 

Introducing  the  new  variable 

4 

j/  =  ±]-,    whence  y  =  ^, 
x  y 


x^—y-  or  xyy'  —y2  is  readily  found  to  be  the  solution. 

rvf 
Eliminating  y'  from  ^<-  =  a  and  xyy'—y-  =  b  gives 

i^ 

^jr2  —  }r  =  ^ 

as  the  solution  of  the  original  equation. 

Compare  this  method  with  that  of  §  27  or  of  §  28,  I',  Remark. 

Ex.  2.   n<"+/2=i. 

Since    x   is    absent,    this    equation    is   invariant   under    Uf= 
(I',  §  28).     Here 


By  either  of  the  methods  of  §§  32  and  33,  the  solution  common  to 
Af—  o  and  (/'/=  o  is  easily  found  to  be/Vi  —}'''• 
Introducing  the  new  variable 


—  y*,  whence  y  = 


§39,40  ORDINARY,   OF  THE   SECOND   ORDER  137 

The  corresponding  ordinary  differential  equation 


has  for  solution  x  —  Vy2—  y12  =  const.     Passing  back  to  the  originnl 
variables  this  becomes  x—  yy'  =  const.     Eliminating  y'  from 


—  y'2  =  a  and  x—  yy' =  l>, 
gives  as  the  solution  of  the  original  equation 

Ex.  3.  y"  =/2  +  i.          Ex.  4.    x?yy"  +  (xy1  -  yf  =  o. 
Solve  examples  of  §  28  by  the  method  of  this  section. 

40.    Differential   Equation   of   the  Second  Order  Invariant  under 
Two  Groups.  —  Since,  if  the  two  groups  6^/and  U<J  leave 

(98)  /'  =  ^(*,J,/) 

unaltered,*  the  corresponding  partial  differential  equation 


is  invariant  under  the  extended  groups  67iyand  &>/,  the  methods  of 
§  38  may  be  employed  to  solve  (98). 

Ex.  1.    xyy"  +  xy'*  -yy'  =  o  (Ex.  i,  §  39). 

This  equation  is  invariant  under  U^f=  x  ~  +7^  an<^  ^/=  y  -•]  " 

,. 


*Then  (98)  is  also  invariant  under  Uf^alUlf-{-a.1lr.1f  (Remark,  §  39).  It  is 
possible  that  Uf  may  assume  simpler  forms  than  U^f  or  £/2/  for  certain  choices  of 
the  constants  a±  and  a.j. 


138 


THEORY  OF  DIFFERENTIAL  EQUATIONS 


A  = 


i   y 


x    y 
o    y 


xy 
o 


=  2     /- 


Hence  the  method  of  B,  4,  (a),  §  38  applies.     The  solution  com- 
mon  to  Af—  o  and  U-^f—  o  is 


I 


dx     dy  dy^ 

1       y      

x      y  o 


=  r  _£+*+*_  ,00*1 

J          ^    +   V   ^    V'  ~       8   ^ 


y     y 


1  A 


The  solution  common  to  Af=  o  and  £72/=  o  is 
fo     ^  dy 

v,     tfj-qh 


J 


=  log 


y'(y  -  x/ y'(y  -  * 


The  general  solution  of  the  original  differential  equation  of  the 

yy' 

second    order    is    found     by    eliminating    y    from     —  =  a    and 


-  =  b  to  be  ax*  —  y2  =  ct  where  c  =  ab. 
'  V 


*  The  method  of  $  32  is  also  available  for  finding  these  common  solutions. 


§40 


ORDINARY,   OF  THE  SECOND   ORDER 


139 


Ex.  2.  /'  =  F(x)y'  +  Q(x)y  +  X(x). 

If  y=\\  and_y=_)'a  are  particular  solutions  of  the  abridged  linear 
equation  /'  =  iy  +  Qyt  the  general  linear    equation    is    invariant 


under  VJmy       and  VJ^y-d  28,  VII,  Note).     Here 
oy  oy 


4+ 

4+ 


ay 


ay 


where  yj  and  j2'  stand  for  -2-1  and  -^2  respectively. 

A  =}'i}'2  —y<>y\  3=-  o,  since   the  two  particular  solutions  are  sup 
posed  to  be  independent. 

( C/i  (/.>')/  =  o.     Hence  method  of  B,  4°,  (a),  §  38  applies. 

Since  y  =  y\  and  y  =y2  are  solutions  of  the  abridged  linear  equa- 

^°n>  ,"_/5,'_|_/9;  A  ,H_p,l  ft 

whence          P 


Introducing  these  values  in  the  expression  for  the  solution  com 
mon  to  Af=  o  and  Ul'/=  o, 

dx     dy 

i    y  jy- 

o      v, 


we  have 
*» 


140  THEORY   OF   DIFFERENTIAL   EQUATIONS  §40 

Noting  that   -j'ij'0'2>i"  -JiW)  =  -}'i"}'&  +Ji>A',  the  first  quad 
rature  is  readily  effected,  thus  giving 

^  =  ^y-j'i>  _  c  _M  ___  ,/JC 

y*y*-y*yi    J  y\y*-y*yi 

Similarly,  the  solution  common  to  Af  •=•  o  and  U»f=  o  is 

^=^y-^v  ..  c    v«*  _^ 

-  J        - 


The  general  solution  of  the  original  equation  is  found  by  eliminat 
ing  y'  from  </>  =  <r2  and  \l/  =  c\  to  be 

j  =  'ij*  +  ^'2  -^i  P—  r^—  ,  ^  +  J2  f-  -r^-7  ^« 

^  j'U's  -  }'2}'i  J  )\yj  -}',}',' 

Note.  —  It  is  an  interesting  fact  that  this  form  of  the  solution  is 
exactly  that  obtained  by  the  method  of  variation  of  parameters 
(El.  Dif.  Eq.  §  49)  from  the  complementary  functionjy=  clyl  +  c2y2, 
as  may  be  easily  verified. 

Ex.  3.    y=7y  +  <2;< 

This  equation  being  homogeneous  in  y,  y',  y",  it  is  invariant  under 

6i/"=7~-(IIIj  §  28).     Moreover,  if  y=yl(x)  is  a  particular  solution, 

df 

the  equation  is  also  invariant  under   (72f=yl^~  (§  28,  VIII,  Note). 

By 

He,<  ^.+ 


f/_  .,     ,  .,  > 

=y\  ,~  -\-y\  -r-,- 

d\-  dy' 

A  =}'}','  -y')\  &  o,   (  6V  <72')/ 
Hence  the  method  of  B,  4°,  (£),  §  38  applies. 


§40  ORDINARY,   OF  THK   SECOND   ORDKK  141 

The    solution    common  to  Af=o  and    U»f=o  is  given  by  that 
method  in  the  form 


AvV  -  PyJ  -  Q  wWx-y,'<ty  +yjy' 

J  **'->& 

Replacing   ^  Vj  by  its  value  y^'  —  P\y\,  this  quadrature  is  readily 
effected,  giving 


A  more  convenient  form  for  the  solution  is 


To  find  a  second  solution  of  Af  =  o,  introduce  the  new  variable 

r  ,  i     \I'dJC 

j  =  *-*""(}>%  -M'),  whence  /=^  +y6  -, 
and     /  =  o  takes  the  form 


dx  y\  dy 

The  corresponding  ordinary  differential  equation 

(iy  _  yj_     _  jfr^ 

dx       r,  "  y\ 

is  linear  with  the  obvious  integrating  factor  —  •     Its  solution  is 

y\ 

v     ,  c^1'1"  * 

' y  \   — -  ax  =  const. 

}'i        J   y\ 

*  Here  y'  appears  as  a  constant,  ($  35). 


142  THEORY  OF  DIFFERENTIAL   EQUATIONS  §§40,41 

The  general  solution  of  the  original  differential  equation  is  found 
by  eliminating  y  from 

v          Cf 
<I>  =  a  and  *-  —  <$  I      —  <L\  =  b 


Cf 
<$  I      —  <L\  = 

J     ' 


to  be  y  =  ay i  I  '  —  dx  -+- 


=  ".''. /%<* 


?.  —  This  is  the  same  form  of  the  solution  as  is  given  by  the 
usual  method  (EL  Dif.  Eg.  §  53,  i°). 

Ex.  40y+y2=i  (Ex.  2,  §39). 

This  equation  is  also  invariant  under   Uf  =  x  ~  4-  y  ;    • 

o.v         o_y 

Ex.5.  y=y2+i. 

Since  jc  and  _y  are  both  absent,  two  available  groups  are 


Ex.  6.  x?yy"  —  (xy*  —  y)~  =  o  (Invariant  under  VI  for  all  values 
of  ni  hence  under  III,  III',  IV,  etc.). 

EX.    7.      (**+/)y'+  2  O'  -*)'')(!  +y2)  =0  (EX.    2,   §    28). 

Ex.  8.    :rv"  -f  A:2j'/2  —  2  xy'  +  2  =  o. 

This  equation  is  invariant  under  I  and  III'. 

41.  Other  Methods  of  Solution.  —  By  making  use  of  the  properties 
of  what  Lie  calls  r-  parameter  groups  of  infinitesimal  transformations 
(§  43)  the  method  of  solving  a  differential  equation  of  the  second 
order  invariant  under  two  groups  can  be  modified  so  as  to  be  con 
siderably  simpler  both  as  to  the  number  of  cases  to  be  distinguished 
and  as  to  the  actual  processes  involved  in  obtaining  the  solution. 
A  brief  study  of  these  groups  will  be  made  in  this  chapter,  leading 
to  the  methods  of  solution  in  §§  46  and  47. 


§42  ORDINARY,   OF  THE   SECOND   oi$DER  143 

42.  Number  of  Linearly  Independent  Infinitesimal  Transforma 
tions,  that  Leave  a  Differential  Equation  of  the  Second  Order  Unaltered, 
Limited.  —  Since  a  differential  equation  of  the  first  order  always  has 
an  integrating  factor,  in  general,  (El.  Dif.  Eq.  §  5)  it  is  left  unaltered 
by  an  indefinite  number  of  infinitesimal  transformations,  the  general 
expression  for  whose  symbols  involves  two  arbitrary  functions  (§  15). 
On  the  other  hand,  a  differential  equation  of  the  second  (or  higher) 
order  is,  in  general,  not  left  unaltered  by  any  infinitesimal  transforma 
tion  (see  Note  IV  of  the  Appendix),  although  some  of  them  are. 
We  shall  prove  the 

THEOREM.  —  A  differential  equation  of  the  second  order  cannot  be 
left  unaltered  />y  more  than  eight  linearly  independent*  infinitesimal 
transformations* 

Suppose  that  the  equation 
(98)  ?  =  F(x,y,f) 

is  invariant  under  the  nine  linearly  independent  infinitesimal  trans 
formations  {/if,  U,f,  ••',  U»f,  it  is  also  invariant  under 


(102)          Uf= 

for  all  possible  choices  of  the  constants  aif  a2,  •••,  ay  (Remark,  §  39). 
It  is  a  well-known  theorem  in  the  Theory  of  Functions  that,  in 
general,  a  unique  integral  curve  of  a  differential  equation  of  the  sec 
ond  order  and  first  degree  (98)  passes  through  two  points,  lying 
within  a  definite  region  determined  by  (98).  Suppose  that  /*,,  /?_,, 
/?!,  /4  in  Fig.  5  are  four  points  such  that  each  of  the  six  pairs  that 
can  be  formed  of  them  determines  a  distinct  integral  curve  of  (98). 
The  nine  constants  ah  a*,  •••,  a9  can  be  so  chosen  that  (102)  leaves 


*  A  set  of  infinitesimal  transformations  f/j  f,  lr%f,  •••,  Urfis  said  to  be  linearly  in 
dependent  if  there  is  no  linear  relation,  with  constant  coefficients,  connecting  their 
symbols;  i.e.  if  it  is  impossible  to  find  a  set  of  constants  <,.  <•._,,  •••,  cr  such  that 


144 


THEORY  OF   DIFFERENTIAL   EQUATIONS 


§42 


each  of  these  four  points  unaltered.  For,  if  their  coordinates  are 
respectively  (xlt  }\),  (x2,y2),  (x3,  ys),  (x4,  y4),  the  requirement  for  this 
is  the  simultaneous  satisfaction  of  the  eight  equations 


(/=  i,  2?  3>  4-) 

These  equations  determine  finite  values  of  the  ratios  of  eight  of  the 
s  to  the  ninth  one  (excepting  possibly  for  peculiar  choices  of  the 

four  points,  which  can  be  avoided) 
because  of  the  linear  independence  of 
the  nine  transformations. 

With  the  tf's  thus  chosen,  the  trans 
formation  (102)  leaves  the  four  points 
Plt  P2,  Ps,  ^4  unaltered  and,  therefore, 
also  the  integral  curves  determined  by 
any  two  of  the  points,  since  integral 
curves  are  transformed  into  integral 
curves  by  a  transformation  which  leaves 


FIG.  5 


a  differential  equation  unaltered,  and  the  four  points  were  so  chosen 
that  through  any  two  of  them  passes  a  unique  integral  curve.  Thus 
through  each  one  of  the  points,  e.g.  through  /*„  pass  three  of  these 
invariant  integral  curves.  The  point  Pl  on  these  being  left  unaltered 
by  (102),  their  slopes  at  this  point,  which  may  be  designated  by  j12', 
.i'i:;'»  Jut  respectively,  are  also  left  unaltered  by  it.  Hence  if  rf  is  the 

coefficient  of  --  in   the   extended   transformation  corresponding  to 
d/ 

(102),  it  follows  that 


for  x  =  xlt  y=yi,  y'=y\*,yv!, yu>     Letting  a,  b,  c  be  the  values 


§42  ORDINARY,   OF  THE   SECOND   ORDER  145 

of  n~>   T*-^>   ~^  respectively  when  x  =  xl9  y=y\,  (103)  gives 
ox     oy      ox         oy 

the  three  relations  a  +  ^,  +  ^,2  =  O} 


a  +  tVu'  +  OW2  =  o. 
Since  the  determinant  of  the  coefficients 


I       >'13       J'lS 

i   ;•»'   V2 

is  different  from  zero,  a  =  b  =  c  =  o.  Hence  if •=  o  for  every  integral 
curve  through  /\,  whence  every  integral  curve  through  Pl  is  invariant* 
under  (102). 

In  exactly  the  same  way  it  can  be  shown  that  every  integral  curve 
through  each  of  the  other  points  PZt  P3,  P+  is  left  unaltered  by 
(102). 

If  P  is  any  fifth  point  in  the  region  containing  Plt  P2,  P3,  P4,  it 
will  lie  upon  at  least  two  f  integral  curves  each  of  which  passes 
through  one  of  those  points.  These  integral  curves  being  invariant, 
the  point  P  is  left  unaltered  by  (102).  In  this  way  every  point 
of  the  plane  (with,  perhaps,  exception  of  certain  points  determined 
by  the  differential  equation)  is  found  to  be  left  unaltered  by  (102). 
The  latter  must  therefore  be  identically  zero  ;  i.e. 

<*i  UJ+  a,  (72f+  •  •  •  +  a9  UJ  =  o. 

*  This  follows  from  the  fact  that  a  unique  integral  curve  of  a  differential  equation 
of  the  second  order  is,  in  general,  determinol  by  tin-  conditions  th.it  it  pass  through 
a  given  point  (x,y)  and  have  a  given  slope  y'  at  that  point. 

f  If  P  does  not  lie  upon  any  of  the  six  integral  curves  determined  by  the  four  points 
(which  is  the  general  case),  this  number  is  four ;  it  is  three  if  P  is  on  one  of  these 
curves,  and  two  if  it  is  at  the  intersection  of  two  of  them. 


146  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§42,43 

Hence  any  nine  infinitesimal  transformations  which  leave  a  differen 
tial  equation  of  the  second  order  unaltered  cannot  be  linearly  inde 
pendent.  This  proves  the  theorem. 

The  differential  equation^1"  =  o  is  a  simple  example  of  an  equation 
that  is  left  unaltered  by  the  maximum  number  of  infinitesimal  trans 
formations.  For,  since  its  integral  curves  are  the  straight  lines  of 
the  plane,  y  =  ax  +  b,  it  is  left  unaltered  by  every  projective  trans 
formation  a4x  +  a:> 


a*y  +  0  9  a7x  -f-  a^y 

In  Note  VI  of  the  Appendix  it  will  be  seen  that  there  are  eight 
linearly  independent  infinitesimal  projective  transformations. 

Remark.  —  In  the  case  of  a  differential  equation  of  higher  order 
than  the  second,  the  following  theorem  holds  :  A  differential  equation 
of  the  nth  order  (n  >  2)  cannot  be  invariant  under  more  than  n  -f  4 
linearly  independent  infinitesimal  transformations.  A  proof  of  this 
theorem  maybe  found  in  Lie,  ContinuUrliche  Gruppcn,  pp.  296-298. 

As  in  the  case  where  ;/  =  2,  a  differential  equation  of  order  //  >  2 
is  in  general  not  left  unaltered  by  any  infinitesimal  transformation. 
On  the  other  hand  the  differential  equation  y(n)  =  o,  «  >  2  is 
invariant  under  each  of  the  n  -f-  4  transformations  (Examples,  §  26) 

df     df       df       df        df       odf  ,df       ,d/  ,   /  N      df 

/  ,  ,;,  *¥->yt->  *1L>  *~  :  {->  '">*    T"  '  ^/  +  ^-1)^^-- 

dx     dy        dx       dy        dy         dy  dy         dx  dy 

43.  r-parameter  Group  of  Infinitesimal  Transformations.  —  Start 
ing  with  a  set  of  infinitesimal  transformations  U^,  U*f,  •••,  Urft  the 
infinitesimal  transformations,  whose  symbols  are  obtained  from  these 
by  applying  the  alternating  process  to  them  in  pairs,  may  or  may  not 
be  linearly  independent  of  them. 

Thus,  if 

the  transformations   (f/i^/,.)/=,    (  Ui 
arc  all  indcjicndcnt  uf  them. 


§43  ORDINARY,    OF   Till';    SFX'OND    ORDKR  147 

On  the  other  hand,  if 

U\  f=  x  -~ ,    £/•>/=  x  „    ,    U?tf^.(x  +  y}  -~- , 
u*  uy  o* 

the  transformation 

(  [/•>  U-£)f=.  x  -• f~  —  (jr  4-  y )  jr~  is  independent, 


while  (tfitfa)/=  tfs/,    (£/itfi)/=E  6V- 

Finally,  if 


none  of  the  new  transformations  are  independent  of  them;   for 


The  case  where  none  of  the  new  transformations  are  linearly  inde 
pendent  of  the  old  ones  is  of  special  interest.  If  r  linearly  inde 
pendent  infinitesimal  transformations  U\ft  U^f,  •••,  Urf  have  the 
property 

(104)  (£W)/=  ^,-1^7+  ^l/2^/+  -  +  ^-,P6;/,  (/,/=  i,  2,  ...  r), 

where  the  «'s  are  constants,  the  aggregate  of  these  and  all  the  trans 
formations  Uf^L  <*\U\f  +  &tU*f  •*?  "•  -\-  aTUTf  where  these  fl's  are 
any  constants  constitute  an  r-parametcr  group  of  infinitesimal  trans 
formations.* 

Remark  i.  —  An  r~  parameter  group  of  infinitesimal  transformations 
is  determined  by  any  r  of  its  transformations  which  are  linearly  inde 
pendent,  since  the  symbols  of  all  its  transformations  can  be  expressed 
linearly  with  constant  coefficients  in  terms  of  any  r  independent  ones. 
Moreover  it  is  readily  seen  that  any  set  of  r  linearly  independent 
transformations  of  the  group  have  the  property  (104). 

*  In  Note  VI  of  the  Appendix  an  r-parumeter  continuous  group  containing  both 
finite  and  infinitesimal  transformations  is  denned.  The  intimate  relation  between  these 
two  classes  of  groups  is  brought  out  in  Lie's  Principal  Theorem  at  the  end  of  the  Note. 


148  TIIKORY   OF   DIFFERENTIAL   EQUATIONS  §43 

Turning  our  attention  now  to  the  'transformations  which  leave  a 
differential  equation  of  the  second  order  unaltered,  we  shall  first  prove 

THEOREM  I.  —  If  a  differential  equation  of  the  second  order  is  in 
variant  under  U\f  and  U^f,  it  is  invariant  under  (U\U»)f. 

For,  if  6^/and  U^f  leave 

(98)  ?  =  F(*,y,f) 

unaltered,  the  extended  transformations  £7^  and  Ujf  leave 


x  ,, 

unaltered.     By  the  theorem  of  §  37,  (£//£/:/)/  or  its  equal 

(see  Note  V  of  the  Appendix)    leaves    Af  =  o   unaltered.     Hence 

Theorem  I  follows.* 

In  §  42  it  was  established  that  the  number  of  linearly  independent 
infinitesimal  transformations  that  leave  any  differential  equation  of 
the  second  order  unaltered  is  limited.  If  in  the  case  of  a  given 
differential  equation  this  number  is  r,  all  the  infinitesimal  transforma 
tions  leaving  the  differential  equation  unaltered  are  linear  functions, 
with  constant  coefficients,  of  any  set  of  r  linearly  independent  ones 
UJ,  #>/,  •••,  Urf.  By  Theorem  I  (£/£/)/,  for  /,/=  i,  2,  •••,  r, 
must  also  leave  the  differential  equation  unaltered.  Hence  they,  too, 
are  linear  functions  with  constant  coefficients  of  the  set  U\ft  U^f,  •••, 
Urf.  The  latter  therefore  have  the  property  (104),  and  we  have  thus 
established 

THEOREM  II.  —  The  aggregate  of  all  the  infinitesimal  transforma 
tions  leaving  a  given  differential  equation  of  the  second  order  unaltered 
constitute  an  r-parameter  group.  Here  o  <><  8.f 

*  This  theorem  is  true  for  a  differential  equation  of  any  order,  and  is  proved  in  the 
same  way. 

f  The  same  theorem  is  true  for  a  differential  equation  of  the  »-th  order,  where  n  >  2. 
In  this  case  o  <  r  <  n  +  4. 


§43  ORDINARY,   OF  THE   SECOND   ORDER  149 

It  is  possible  that  a  smaller  number  than  r,  say  s,  of  linearly  inde 
pendent  infinitesimal  transformations  in  an  r-parameter  group  deter 
mine  a  group  ;  the  latter  is  known  as  an  ^-parameter  subgroup  of  the 
larger  group. 

The  four  transformations 


determine  a  four-parameter  group  ;    for  they  are  linearly  independent,  and  besides 


(UzU*)f=Utf-  L\f,   (U*Udf=-U* 

Of  the  subgroups  of  the  four-parameter  group  the  following  are  immediately 
obvious  : 

The  two-parameter  subgroups  i7\f,  U»f\  U\f,  U,\f;  U\f,  U\f\  U^f^  U\f\ 
Uzft  U*f. 

The  three-parameter  subgroups  L'\/,  U»f,  U\f\    U\f,  L  ':{/,  L\f. 

Uf^x^-—y^L=U\f—  U\f,  also  a  transformation  of  the  four-parameter 
group,  determines  with  U^f  and  U$f  a  three-parameter  subgroup,  since 

=  2  U-,f,    (  U.£  6s)/=  -  Uf. 


Remark  2.  —  Starting  with  two  or  more  linearly  independent  infini 
tesimal  transformations  which  leave  a  given  differential  equation  of 
the  second  (or  higher)  order  unaltered,  a  group  of  infinitesimal  trans 
formations  is  determined  which  is  either  the  r-parameter  group  of 
Theorem  II  or  a  suogroup  of  it. 

For,  let  £/i/,  £/2/,  •••,  UJ,  (2<k<r)  be  a  set  of  linearly  inde 
pendent  transformations  which  leave  the  differential  equation  of  the 
second  order  unaltered.  By  Theorem  I,  (£/|£/)/,  (/,/=  i,  2,  •••,  X') 
also  leave  the  differential  equation  unaltered.  Some  or  all  of  thq^e 
may  be  linearly  independent  of  the  original  ones.  Let  /:'  of  tin  in 
be  such.  We  know  that  k  +  k'  <  8.  Adding  these  to  the  original  set, 
combine  the  larger  set  in  pairs  by  the  alternating  process  as  before. 


150  THEORY   OF   DIFFERENTIAL   EQUATIONS  §43 

The  resulting  transformations  also  leave  the  differential  equation  un 
altered.  If  any  of  these  are  independent  of  the  members  of  the 
larger  set,  add  them  to  the  latter,  thus  forming  a  still  larger  set  of 
linearly  independent  transformations  leaving  the  differential  equation 
unaltered.  Proceed  with  this  set  as  before.  Obviously  this  process 
must  be  a  finite  one,  since  the  maximum  number  of  members  of  a 
set  is  eight.  So  that  the  above  process  stops  when  no  new  trans 
formations  independent  of  the  previous  ones  arise  as  a  result  of  the 
alternating  process.  If  the  number  of  independent  transformations 
finally  appearing  is  r,  the  r-parameter  group  determined  by  them  is 
precisely  that  of  Theorem  II  ;  if  the  number  is  s  <  r,  the  ^--parameter 
group  determined  by  them  is  a  subgroup  of  the  other. 
We  shall  prove 

THEOREM  III.  —  Every  r-parametcr  group  (r  >  2)  contains  tivo- 
paramcter  subgroups.* 

As  a  matter  of  fact  we  shall  show  that,  fixing  upon  any  one  of  the 
transformations,  say  UJ~,  a  set  of  r  —  i  constants  c.,,  c3)  -••,  cr  can  be 
found  such  that 


constitute  a  two-parameter  subgroup  ;  it  being  understood  that  the 
r-parameter  group  is  determined  by  6^/,  U,f,  •••  Urf,  which  are, 
therefore,  subject  to  the  conditions 

r  t 

(104)  (Utf)f=  ^a,JkUkf,    (ij=  i,  2,  .-.,  r). 

k  i 

In  order  that  this  be  the  case 

(fsV«/+  c,UJ 


-f  -.  + 

*  This  theorem  and  its  proof  hold,  without   modification-,  for  groups   involving  n 

variables. 


§43 


ORDINARY,   OF  THE   SECOND   ORDER 


Since 


(U»          M\fmc£UJWmc^aj^    and    since 


U\f,  UJ,  "•,  Urf  are  linearly  independent,   (105)  can  hold  only  in 
case 


(106) 


=  a 


=  2,  3,  ~; 


j=2 


Conversely,  if  c^  <r3,  •••,  cr  can  be  found  to  satisfy  r  equations  of  the 
type  (106),  where  a  and  b  are  any  constants,  and  not  all  of  the  <:'s 

zero,  the  group  Uf=  jVj&^/will  determine  with  UJ  a  two-param- 
13 

eter  subgroup  ;  for  in  this  case 


That  such  a  set  of  c's  can  always  be  found  may  be  seen  as  follows : 

The  last  r—  i  equations  of  (106)  are  the  linear  homogeneous  equa 
tions 


(107) 


These  can  be  solved  provided  b  satisfies  the  equation 


(108) 


=  0. 


This  equation  necessarily  contains  b,  since  the  coefficient  of  br~^  is 
(—  i)1""1.  Using  any  value  of  />  satisfying  it,  the  c\  are  determined 
to  within  a  common  factor  (which  is  not  essential),  by  solving  (107). 
The  value  of  a  is  then  determined  by  the  first  equation  of  (106). 
Thus  Theorem  III  is  not  only  proved,  but  a  method  for  finding  the 
two-parameter  subgroup  is  also  given. 


152  THEORY   OF   DIFFERENTIAL   EQUATIONS  §§43,44 

The  transformations 

w=f,  *v=f^  u*f=y&+*y: 

dx  dy  dx        dy 

determine  a  three-parameter  group,  since 


Inspection  shows  that  £/]/and  U»f  determine  a  two-parameter  subgroup.  To 
find  another  two-parameter  subgroup  of  which  Us/is  one  of  the  determining  ele 
ments,  the  method  of  this  section  may  be  employed.  The  constants  c\  and  c% 
must  be  so  determined  that 


c,  £/»)/=  a  U-,/+  6(fl  Ui/+  c2  6V)  > 
i.e.  -  fi  6V-  c»  Uif=  a  U*f+  bc^  6V  +  be*  Uzf. 

.•.  a  =  o,    bc\  +  c»  =  o,    bc»  +  c\  =  o. 

In  order  that  the  last  two  equations  be  consistent,  b  must  satisfy  the  equation 
ft*  —1=0, 

whence  b  =  .  ±  I  and  —  =  T  I.     Hence 
c-i 

*  =  ,&  +  ,%,  Uf=&+%  and  U^,  &+JC&,Ufs&-& 

d*       dy  dx     dy  dx        dy  dx     dy 

are  two  two-parameter  subgroups  of  the  original  group. 

44.    Classification  of  Two-parameter  Groups.  —  If  a  two-parameter 
group  is  determined  by  £/L/and  U*f, 


Either  both  al  and  a2  are  zero  or  they  are  not.  In  the  latter  case  it 
is  always  possible  to  find  a  pair  of  transformations  to  determine  the 
two-parameter  group  for  which  one  of  these  constants  is  unity  and 
the  other  zero.  For,  if  #,  ^=  o 


§44  ORDINARY,   OF  THE   SECOND   ORDER  153 

are  linearly  independent  and 


Moreover,  if  for  any  pair  of  linearly  independent  transformations 
and  U-J  of  a  two-parameter  group 


this  is  true  for  every  pair,  since 

=  o 


for  all  choices  of  constants  clt  c2,  b^  b^.     Hence  every  two-parameter 
group  can  be  represented  by  a  pair  of  transformations  U^f  and 

such  that  either 

=  o  or  (  UtUftfm  UJ. 


These  two  possibilities  are  mutually  exclusive ;  any  grotip  can  come 

under  one  head  only. 

A  second  mode  of  classification  is  suggested  by  the  following : 
If  a  two-parameter  group  is  determined  by  U^f  and  U<J  which  are 

connected  by  a  relation  of  the  form 

where  p(x,  y)  is  not  a  constant, J  every  pair  of  distinct  transforma 
tions  of  the  group  are  connected  by  a  relation  of  the  form  (109)  ; 

*  If  «!  =  o,  rt.j  ^fc  o,  so  that  ( Ul  &y/=  a^U^f,  the  groups 


satisfy  the  condition  (  V\  K2)/=  V±f. 

t  It  is  interesting  to  note  that  (£/i  £/;>)/=  o  is  the  necessary  and  sufficient  condition 
that  each  transformation  of  the  group  generated  by  U^  f  be  commutative  with  every 
transformation  of  the  group  generated  by  £/2/  For  an  elementary  proof  of  this  fact, 
see  Lie,  DiffcrfntuilglciclniHgen,  p.  305. 

t  While  in  this  case  f'j/and  ('.,/  are  distinct  transformations,  the  one-parameter 
continuous  groups  generated  by  them  have  the  same  path-curves. 


154  THEORY   OF  DIFFERENTIAL   EQUATIONS  §44 

for  if 


Hence  all  two-parameter  groups  may  be  divided  into  two  classes 
according  as  their  distinct  transformations  are  connected  by  a  relation 
of  the  form  (109)  or  not. 

These  two  modes  of  classification  are  independent  of  each  other. 
Hence  four  classes  of  two-parameter  groups  may  be  distinguished  ac 
cording  as  thc\  are  representable  by  a  pair  of  transformations  U^f  and 


U,f=p(x,y}L\ft 
P  (x,  r) 


Classify  the  following  two-parameter  groups  : 

£  +  (;-x)Z      xf+y 

dx  dy        dx        d 


Ex.   3.    x(x+y)+y(X+y),    X(X  -y)      .+;(*  -  v)       . 


OUT  OV 

.   5.     -  —  , 


x.  6.    .v       +  v, 

f).v      '  ay 


.v+;' 


§45  ORDINARY,   OF  THE   SECOND   ORDER  155 

45.  Canonical  Forms  of  Two-parameter  Groups.  —  By  a  proper 
choice  of  variables  the  various  classes  of  two-parameter  groups  can 
be  reduced  to  certain  simple  forms  which  Lie  called  their  canonical 
forms.  These  will  now  be  determined  in  turn. 


By  the  method  of  §  9  a  set  of  variables  can  be  determined  so  that 
Uif  takes  the  form 


dx 
If  the  resulting  form  of  U^f  is 


^EEO  and   ^  =  o,  since    (U,U*)f=  ^  Bf-  +  ^  ¥  =  0.       Hence 

ox  ox  dx  ox      ox  oy 

£  and  77  are,  at  most,  functions  of  y  ;  /'.<?. 


where  rj(y)  =£  o,  since  U,f^  p(x,  y)  UJ. 

The  transformation  ^/,/=  -^   remains  unaltered  by  a  change  of 


variables  of  the  type 


where  <f>(y)  and  ^(j)  are  at  our  disposal.     This  change  of  variables 
causes  £/>/to  take  the  form  [(15)  §  9] 


assumes  the  form        . 


If  <#>  =  f^  dy  and  ^(  v)  =  f  ^'    ,   UJ 
J  iy(>')  »/  T;(J) 


156  THEORY   OF   DIFFERENTIAL   EQUATIONS  §45 

Hence,  by  a  proper  choice  of  variables,  a  pair  of  transformations 
satisfying  conditions  a  can  be  made  to  assume  the  canonical  forms 


ox  dy 

Having  established  the  existence  of  the  canonical  forms  in  this 
case,  the  actual  finding  of  the  canonical  variables  (which  reduce  the 
transformations  to  these  forms)  can  be  accomplished  by  two  quadra 
tures.  For,  starting  with  a  pair  of  groups 


+  and 

ox          oy  ox          oy 

satisfying  conditions  a,  the  new  variables  x  and  y  will  reduce  these 
to  the  forms  ^  /•  •)  /• 

UJ=%-  and   IV*? 
dx  dy 

respectively,  provided  x  satisfies 

f.  dx  ,       dx  f.  dx  ,       dx 


and  y  satisfies      fi  |»  +  „  ^  =  o,   (,  f-  +  „  ?»  =  ,. 

dx          dy  dx          dy 

Since  such  new  variables  must  exist,  the  equations  of  each  pair 
must  be  consistent.     Because   Utf^pU\ft  they  can  be  solved  for 

—  t  JL  and  —  ,  —  respectively,  whence  x  and  y  are  determined 
dx     dy  dx     dy 

by  the  two  quadratures 


x  ay 

The  transformations 


§45  ORDINARY,   OF  THE   SECOND   ORDER  157 

form  a  pair  satisfying  conditions  (t.     From 

-  +  ,•:=,  and  +*=o 


whence  x 

Similarly,  from    _>!  &  +  *  |*  =  o  and  *  ft  +,  f*~  I, 

d*         07  d-r     '  dj' 

5£=—  ^—   and    ^^ 

^ 


whence  y  -  =  }      V 

J     «•  +  y2 

These  canonical  variables  are  obvious  from  geometrical  considerations. 


As  before,  £/J/"can  be  reduced  to  the  form 


by  the  choice  of  canonical  variables  (§  10).     Then  t/,/  assumes  the 
form  ., 


where  O-(AT,  y)  is  what  p  becomes  when  the  old  variables  are  replaced 
by  the  new.     Since 


dy  dy 

cr  is  a  function  of  x  only.     Taking  this  as  the  new  A*,  which  change 

of  variables  leaves  U\f  unaltered,  £7>/assumes  the  form  .v  -    •     Hence, 

ov 


by  a  proper  choice  of  rtiriiifi/t's  (7  pair  c/  transformations  satisfying 


158  THEORY   OF   DIFFERENTIAL   EQUATIONS  §45 

conditions  fi  can  be  made  to  assume  the  canonical  forms 

TTf-ty      .T,_      d/ 

v\J  =  T-  ,    c  o/  =  A  -  -  • 
3;'  dy 

The  actual  finding  of  the  canonical  variables  in  this  case  requires 
a  single  quadrature.     For,  starting  with  a  pair  of  transformations 

£V=*£  +  ,£  and    ^/Wf^  +  JA 

ax         dy  \~  ox         vyj 

where  (  L\  U^f  =  UlP  UJ  =  o,  i.e.  UiP  =  o, 

the  new  variables  x  and  y  will  reduce  these  to  the  forms 


respectively,  if  x  =  p  (.Y,  v) 

and  y  satisfies  the  equation  (§  10), 

4+^=i. 

fl.v         3>» 
Moreover,  since  U}p  =  o,  p  is  a  solution  of 


Hence,  y  =  F(x,  }')  is  some  solution  of 

dx  _  dy  _  dy 

7~7~T 

distinct  from  p  =  const.,  which  is  also  a  solution  of  this  system  of 
equations.  Among  the  various  ways  that  will  suggest  themselves 
when  £  and  r)  are  given  in  any  specific  case,  a  possible  method  is  to 
solve  p(.\-,  jr)  =  c  for  one  of  the  variables,  say  x  =  <}>()>,  c),  whence 


y 


=  f   <*y 

J  r,(<t>,)>) 


§45  ORDINARY,   OF  THE   SECOND   ORDER  159 

The  transformations 

Uif=xy  &.  +  2y*  V-  and  U->f=x*  ^L  +  2*2y  ^L 

fa  dy  dz  dy 

form  a  pair  satisfying  conditions  p. 


y  must  satisfy  xy^>  +  2y---y-  =  i. 

d*  dy 

To  solve  the  corresponding  system  of  ordinary  equations 

<Xr       dy-  _  dy 
xy      2y*       I 

use  may  be  made  of  the  solution  —  =  c  or  y  =  —  •     Then 

y  c 


2x  2y 

y.  (UMV=  (/i/,   U.J=£p(xt 

As  before,  6^/can  be  put  in  the  form 


by  introducing  canonical  variables.     Taking  6^/in  the  form 


, 
we  must  have  ^r  —  °  an<l  ^   =  J>  since 


Hence,  i^/must  liave  the  form 


160  THEORY   OF   DIFFERENTIAL   EQUATIONS 


where  £(x)  ^  o,  because  U.^f^pU^f.     The  introduction  of  the  new 

variables 

x=t(x)tV  =  il,(x)+y 

leaves  U^f=  —  unchanged  in  form,  but  changes  cZ/into 


d 


dx  dy 


This  takes  the  form  Uo/=  x-~  +y^- 

dx        By 

tdjc 

when  £(x)$(x)  =  <j>(x),  or  <j>(x)  =  e]*W 

r_ 
and  £(*}$'  (x)  +X(x)  =t(x),  or  if/(x)  =  -e^Mj  ^  e~^(^  a'x. 

Hence,  by  a  proper  choice  of  variables  a  pair  of  transformations 
satisfying  conditions  y  can  be  made  to  assume  the  canonical  forms 


The  actual  finding  of  the  canonical  variables  in  this  case  requires 
two  quadratures.     For,  starting  with  a  pair  of  groups 


satisfying  the   conditions  y,  the  new  variables  x  and  y  will  reduce 
these  to  the  forms 


respectively,  provided  x  satisfies 

t  dx  .       dx  t  dx  .       dx 

£15-  +  *?!^  =°>   ^T-  +  ^-- 

ox         dy  dx          dy 


§45  ORDINARY,   OF  THE   SECOND   ORDER  l6l 

and  y  satisfies 

£  dy          dy_        fdy          dy  _ 

ClT--r  »/lT-=  I?      C2-T-  -I-  >?2T-  —  U- 

dx         dy  dx         dy 

Since  such  new  variables  must  exist,  the  equations  of  each  pair  must 

be  consistent.     Because  U-2/^pU\f  they  can  be  solved  for  —  •,   — 
-»        a  dx     dy 

,    dy     dy 

and    -^,   -^,  giving 
dx     dy 


__  ft 


djt       ^772  —  ^r/!         ^172  —  ^i       ^J       ^1^2  —  £2171          Utyi  —  tL-^i 

3  log  x       j    d  log  JT 

Dividing  (no)  by  JT,  -  «•-   and   -  «  —  are  given,  whence  log  JT  is 
ox  dy 

obtained  by  a  quadrature  and  the  form  for  x  follows. 

Equations  (in)  maybe  solved  in  various  ways.  The  most  gen 
eral  form  for  y  satisfying  them  is  not  needed.  As  a  matter  of  fact, 
the  simpler  the  form  obtainable,  the  better.  One  way  of  proceeding* 
is  to  assume  that  x  and  y  are  no  longer  independent,  but  that  y  =  ex 
where  c  is  a  constant.  Then 

(112)  ^=^+^  =  (X  +  ^)i/  +  ,+  ^ 

ax      ox        oy 

where  X,  /x,  v,  IT  are  what  the  corresponding  coefficients  in  (in) 
become  when  y  is  replaced  by  ex.  Since  (112)  is  a  linear  ordinary 
differential  equation  of  the  first  order,  it  may  be  solved  by  the  usual 
method,  involving  two  quadratures  (EJ.  Dif.  /£/.  $j  13).  A  process, 
however,  by  which  a  single  quadrature  alone  is  involved  in  solving 
(112)  is  given  by  the  following  : 

*  Special  methods  will  frequently  be  found  simpler,  however. 


1  62  THEORY   OF   DIFFERENTIAL   EQUATIONS  §45 

Inspection  of  equations  (no)  and  (in)  shows  that  y=x(x,  ex), 
which  is  obtained  from  x(x,  y)  by  replacing  y  by  cx9  satisfies  the 
equation 


Hence  the  transformation  y  =  vx(x,  ex)  reduces  (112)  to 

dv  _  v  -f-  CTT 
dx     x(x,  ex)' 

whence  v  is  obtained  by  a  quadrature.     Then  y  follows  at  once,  after 

replacing  c  by  -L. 

x 

The  transformations 


and  U»f=x'i       +  (y  +  xy) 
dy  dx  dy 


form  a  pair  satisfying  the  conditions  7.     From 


=0  and  x*-\- 
dy  dx  dy 


. 
**'       dy 


=  i 


From  x&=i  and  ^  ^  +  (y  +  xy)       m  y, 

dy  dx  dy 

fa  -  y.  -.y^rj^L    ^-i.* 

dx     x*        *J    '    dy     x 

Putting^  =  cx  -7~^i~~i 

dx      x1      x1 

*  These  equations  can  be  solved  directly.     From   the   second  one  y=- 

where  0(.r)  is  to  be  determined.     Putting  this  value  of  y  in  the  first  equation  gives 

_\ 

4&_£, whence* •*    *. 

dx      x* 

k  =  Q  gives  the  form  for  y  obtained  in  the  text. 


§45  ORDINARY,    OK   TIIK    SWOND    OKDI.R  163 

1 

Using  the  method  given  above,  put  y  —  ve    x.     The  linear  equation  then  reduces 

i  i 

to  —  =  — —  ex  ;   whence  v  —  cex. 

(/JC  X 


8. 


As  before,  by  the  introduction  of  canonical  variables   £/,/  can  be 
made  to  assume  the  form 


These  variables  will  cause  £72/to  assume  the  form 


Since  (  £/,  «,)/  =  UJ,       =  i  and  r,  =  A'(.v)  +y.     So  that 


The  change  of  variables  y  =  X(x)  +  v  leaves   ^/unaltered  and 
changes  U*f  to  the  form  U.,/  =  y~-'     Hence,  f>y  a  proper  choice  of 

variables  a  pair  of  transformations  satisfying  conditions  8   can    /><- 
made  to  assume  tJie  canonical  forms 


The  actual  finding  of  the  canonical  variables  in  this  case  requires 
the  solving  of  the  differential  equation  of  the  first  order  determining 
the  path-curves  of  the  group  generated  by  either  of  the  transforma- 


164  THEORY   OF   DIFFERENTIAL   EQUATIONS  §45 

tions.     For,  starting  with  the  pair  of  transformations, 


where  (U^U^f  =  L\f,  the  new  variables  x  and  y  will  reduce  these  to 
the  forms  .,,  .  , 


respectively,  if  y  —  />  (  JP,  j)  * 

and  x  satisfies  the  equation 

.dx        dx 
^+7^=°- 

The  solution  of  this  equation  is  usually  obtained  by  first  solving 

dx      dy 

T"T' 

the  differential  equation  of  the  path-curves  of  £/,/. 
The  transformations 

,T/^_4/anW-|_,|' 

form  a  pair  satisfying  conditions  5. 


y 

The  solution  of  ^  +  f£/  =  oisi  +  -  =  const. 


Ex.     Determine  the  canonical  variables  for  the  groups  at  the  end 
of  §  44. 

*  The  other  requirement  of  y,  viz.  Uy=  i,  follows  from  the  given  conditions  on  U^f 
and  i/a/  since  (U^ 


§46  ORDINARY,   OF  THE   SECOND   ORDER  165 

46.  Differential  Equation  of  the  Second  Order  Invariant  under  Two 
Groups.  —  Starting  with  two  non-trivial  infinitesimal  transformations  * 
which  leave  a  differential  equation  of  the  second  order 


unaltered,  an  .r-  parameter  group  of  infinitesimal  transformations, 
leaving  (98)  unaltered,  can  be  found  (Remark  2,  §  43),  which  con 
tains  a  two-parameter  subgroup  (Theorem  III,  §  43)  determined  by  a 
pair  of  transformations  ^/and  U2f  which  satisfy  one  and  only  one 
of  the  conditions  (§  44), 

(l/M)/=o  and  (UM)/=  UJ. 

Moreover,  these  two  transformations  can  be  found  by  direct  and 
practicable  processes  from  the  original  two  transformations,  and  they 
also  leave  the  differential  equation  (98)  unaltered. 

We  shall  now  suppose  that  we  have  found  such  a  pair  of  infini 
tesimal  transformations  U\f  and  U^f.  Passing,  as  was  done  in  §  39, 
to  the  corresponding  linear  partial  differential  equation 


the  latter  is  invariant  under  the  extended  transformations  U\J  and 
UJf,  which  are  subject  to  one  of  the  conditions 

(£//£/,')/  EEO  and  (^'6/a')/ 

since  (f//c7/)/=(^c72)7  (see  Note  V  of  the  Appendix).     Two  im 
portant  cases  are  to  be  distinguished  : 
A.    A  relation  of  the  form 

(970  UJf^aWf  +  pAf 

*  As  use  is  to  be  made  of  the  properties  of  groups  of  infinitesimal  transformations, 
the  one-parameter  groups  under  which  (98)  is  invariant  will  be  replaced  by  their  repre 
sentative  infinitesimal  transformations  in  what  follows.  (Compare  Remark,  §  6.) 


1 66 


THEORY   OF   DIFFERENTIAL   EQUATIONS 


§46 


exists.      In  this  case  UJ  and   cZ/  determine   distinct  path-curves, 
that  is,  no  relation  of  the  form 

(109) 


can  connect  them.     For,  if  such  a  relation  did  exist,  and  if 


Uzf  would  have  the  form 


f          do-   .       do-    f      >  d<r   ,       .do-    ,,\3/ 

+*-;£+**;,>'  -feV^'-^   £• 


A  relation  of  the  form  (97')  implies  the  vanishing  of  the  determinant 


—  , 

dx      \'  By 


do- 
dy 


This  reduces  at  once  to 


c)cr        ,.   do-\    , 

^  r*~  ^  T~  / 

^          5^/ 


,.    d(T    ,.,"1 

•*•  «  ^-/ 

qy 


Since  neither  171  —  ^7'  nor  both  ^  and  ^  can  be  zero   identically, 

A  can  vanish  identically  only  in  case  ~  =  o  and  -—  =  o  simultane- 

O.T  dy 


ously,  that  is,  <r  must  be  a  constant.  This  would  make  U\f  and 
one  and  the  same  transformation.  Hence  the  relation  (109)  cannot 
hold  when  (97')  does.  Assuming  that  (97')  holds,  two  cases  must  still 
be  considered  : 


§46 


ORDINARY,   OF  THE   SECOND   ORDER 


I67 


By  means  of  two  quadratures  (§  45,  «)  canonical  variables  can  be 
found  so  as  to  reduce  the  two  transformations  to  the  forms 


dx 


- 

By 


respectively.  Since  the  differential  equation  expressed  in  terms  of 
these  variables  must  be  left  unaltered  by  these  two  transformations,  it 
must  be  free  of  both  x  and  y  (I  and  I',  §  28).  Hence  it  has  the  form 


and  the  corresponding  partial  differential  equation  has  the  form 


loreover,.        _„  _ ^    _  _^ 
The  relation  (97')  implies  that 


i   y 

I       O 
O       I 


=  F(y')  =  o. 


Hence  when  conditions  (97')  and  i°  hold,  the  introduction  of  canoni 
cal  variables  for  the  two-parameter  group   reduces   the  differential 

y"  =  o, 

y  =  ax  -f  /;. 


equation  to  tJie  fonn 
and  the  solution  is 


By  means  of  two  quadratures  (§  45,  y)  canonical  variables  can  be 
found  in  this  case,  reducing  the  transformations  to  the  forms 


dx 


1 68 


THEORY   OF  DIFFERENTIAL   EQUATIONS 


§46 


The  introduction  of  these  variables  reduces  the  differential  equa 
tion  to  the  form  (I  and  IV,  §  28) 

',  xy")=o,  or  y"= 


The  corresponding  linear  partial  differential  equation  has  the  form 


Moreover, 


The  relation  (97')  implies  that 
,     y' 


O       I 

x    y 


EE  -  F(y')  =  o. 


Hence,  also,  in  the  case  where  conditions  (97')  and  2°  hold,  the 
introduction  of  canonical  variables  for  the  two-parameter  group 
reduces  the  differential  equation  to  the  form. 

y"  =  o 

and  the  solution  is  y  =  ax  -+-  b. 

B.    No  relation  of  the  type  (97')  exists.     That  is, 


Here  the  two  subcases  in  A  are  also  to  be  considered. 


Since  this  carries  with  it 


§§  46,  47 


ORDINARY,   OF  THE   SECOND   ORDER 


I69 


the  conditions  of  §  38,  B,  4°,(#)  exist,  and  the  two  solutions  of  the 
corresponding  linear  partial  differential  equation  are  given  by  the 
two  quadratures 


dx     dy 

i    y 
£1     >?i 


where 


AEE 


=  #  and 


/ 


I 


y 


Eliminating  7'  from  these  gives  the  solution  of  the  original  differen 
tial  equation. 


Since  this  carries  with  it 


the  conditions  of  §  38,  B,  4°,  (<£)  exist.  Two  solutions  of  the  cor 
responding  linear  partial  differential  equation  are  obtained  by  two 
quadratures,  by  the  method  given  there.  Eliminating  y'  from  these, 
the  solution  of  the  original  differential  equation  follows. 

Remark.  -•-  It  may  be  noted  that  in  every  instance  where  an 
ordinary  differential  equation  of  the  second  order  is  known  to  be 
invariant  under  two  distinct  groups,  of  which  neither  is  trivial,  its 
integration  can  be  effected  by  means  of  two  quadratures. 

47.  Second  Method  of  Solution  for  B.  —  The  method  in  cases  A, 
i°  and  2°  of  the  previous  section  leaves  nothing  to  be  desired.  For 
the  remaining  cases,  however,  while,  theoretically,  the  reduction  of 
the  problem  to  two  quadratures  seems  sufficiently  simple,  a  method 
analogous  to  that  employed  for  A,  even  if  involving  a  larger  number 


1  70  THEORY   OF   DIFFERENTIAL   EQUATIONS  §47 

of  quadratures,  or  possibly  the  solution  of  a  differential  equation  of 
the  first  order,  may  prove  simpler  in  actual  practice.  Still  under  the 
supposition  B,  viz. 


the  four  possible  forms  (§  44)  of  the  two-parameter  groups  of  infini 
tesimal  transformations  leaving  the  differential  equation  unaltered 
will  be  considered  : 


By  a  process  involving  two  quadratures  (§  45,  a)  canonical  varia 
bles  x  and  y  can  be  found,  reducing  the  infinitesimal  transformations 
to  the  forms 


The    differential    equation   invariant   under   these    has    the    form 
(I  and  I',  §  28) 


An  additional  quadrature  gives 

/rfy' 
F(y') 

or,  when  solved  for  y',      y'  =  <j>(x  -f  #), 
and  a  final  quadrature  gives  the  solution 

y  =   I  (f)  ( x  -f-  (7  )  tix  -f  /^. 

In  this  case  four  quadratures  are  required. 

/,)/EEo,    U,/=P(.r,y)tyj. 


§47  ORDINARY,    OF   TI1IO    SKCOND    ORDFR  I/I 

By  a  process  involving  one  quadrature  (§  45,  /j)  canonical  varia 
bles  x  and  y  can  be  found,  reducing  the  infinitesimal  transformations 
to  the  forms 


The    differential    equation    invariant    under   these    has    the    form 
and  VII,  J  ,8)  yll  =  F(^ 

Two  additional  quadratures  give  the  solution 

y  =f  CF(X)  dx-  +  ax  +  b. 
In  this  case  three  quadratures  are  required. 


By  a  process  involving  two  quadratures  (§  45,  y)  canonical  varia 
bles  x  and  y  can  be  found,  reducing  the  infinitesimal  transformations 
to  the  forms 


The  differential  equation  invariant  under  these  has  the  form  (I  and 
IV,  §  28) 


As  in  the  case  a,  two  additional  quadratures  give  the  solution. 
In  this  case  four  quadratures  are  required. 

8.  (UMV^Ui/,    6i/=PCv,v)^i/ 

By  a  process  (§  45,  8)  involving  the  finding  of  the  path-curves  de 
termined  by  either  infinitesimal  transformation,  i.e.  the  solution  of 
the  differential  equation  . 


1/2  THEORY   OF   DIFFERENTIAL   EQUATIONS  §47 

canonical  variables  x  and  y  can  be  found,  reducing  the  transforma 
tions  to  the  forms 


The  differential  equation  invariant  under  these  has  the  form  (I  and 
HI,  §  28) 


Two  quadratures  give  the  solution 


Remark.  —  The  above  classification  holds  equally  well  for  A,  for 
which  it  is  exceedingly  simple,  cases  ft  and  8  never  arising  (§  46). 
Hence  the  method  of  introducing  canonical  variables  applies  to  all 
cases  where  a  differential  equation  of  the  second  order  is  invariant 
under  two  groups.  The  interest  in  §  46  lies  in  the  fact  that  it  is  there 
shown  that  it  is  always  possible,  if  desirable,  to  solve  the  differential 
equation  by  two  quadratures  only. 

While  the  classification  of  §  40  is  more  complicated,  it  must  be 
borne  in  mind  that  the  two  groups  employed  there  need  not  deter 
mine  a  two-parameter  group.  Some  of  the  methods  of  §  40  are  ex 
ceedingly  simple  ;  so  that  they  are  not  to  be  ignored.  On  the  other 
hand,  it  is  suggested  that  the  method  of  this  section  be  applied  to  the 
examples  of  §  40. 

Ex.  1.    xyy"  +  xyn-yy'  =  o.      (Ex.  i,  §  40). 

This  equation  is  invariant  under  U\f=  x-J-  -f-  y  •£•  and  £Z/=  v    -• 

dx         oy  '  dv 

These  determine  a  two-parameter  group  of  the  type  «.     The  canoni 
cal  variables  are  readily  found  to  be  x  =  log*,  y  =  log^.     Introduc- 

X 

ing  these,  the  differential  equation  takes  the  form 


§47  ORDINARY,   OF   THE   SECOND   ORDER  1 73 

Integrating  this,  one  obtains 

log  -~^  +  '*  =  '°r  if  =  T^F— :  =  ~^»' 


Integrating  again, 

2  y  +  c  =  log  (^  -  e  >2X)  or  ^  =  ^  -  «T2jr. 

Passing  back  to  the  original  variables^ 

^2-^/  =  i. 
Ex.   2.   /'  =  /y  +  Qy  +  X.     (Ex.  2,  §  40)  . 

This  equation  is  invariant  under  £7i/=  V\—  and  K/syf-^,  if 

5y  "  dy 

j/'  =  /'v/  +  <27i   and  >-,/'  =  Pyi  -f  fe-     The  transformations  UJ 
and  ^/determine  a  two-parameter  group  of  type  /3.     The  canonical 

variables  are  x  =  ^,  y  =  2-.     To  introduce  these  use  should  be  made 
* 


of  the  fact  that       =  =      -   where  A  =  \\  y./  -  y2ylfj  and  that 

^r      </or  dx      y\2 

y\y*  '-  -y*yi  '  =  /^.    Then 


Substituting  these  values  in  the  differential  equation  gives 


where  the  right-hand  member  must  be  expressed  as  a  function  of  x. 
Integrating  twice, 


174  THEORY  OF  DIFFERENTIAL   EQUATIONS  §47 

Passing  back  to  the  original  variables, 


t>)  f 

iA' 


, 

(AX*  -jyi  ) 


.  —  It  is  an  interesting  fact  that  this  form  of  the  solution 
includes  as  a  special  case  the  form  obtained  by  a  well-known  method 
in  case  the  coefficients  in  the  linear  equation  are  constants.  (See 
EL  Dif.  Eq.  §  47-) 

Ex.  3.  /'  =  //  +  Qy.     (Ex.  3,  §  40.) 


This  equation  is  invariant  under  U^f=)\--  if  y1  is  a  particular 


solution  of  the  equation,  and   also   under  R/Byu.     The    trans- 

dy 
formations  C^/and  U*f  determine  a  two-parameter  group  of  type  8. 

The  canonical  variables  are  x  =  x,  y  =  2-.     This  change  of  variables 

y\ 

is  the  one  usually  employed.     (See  EL  Dif.  Eq.  §  53,  i°.) 


CHAPTER   VII 
CONTACT  TRANSFORMATIONS 

48.  Union  of  Elements.  —  The  configuration  consisting  of  a 
point  and  a  line  *  through  it  is  known  as  a  lineal  clement.  It  is 
obviously  self-dualistic.  Since  a  lineal  element  in  the  plane  is  deter 
mined  by  three  coordinates,!  there  are  oo  ;>>  such  elements. 

Any  curve  in  the  plane  determines  co  J  lineal  elements,  each  one 
consisting  of  a  point  of  the  curve  and  the  tangent  line  at  that  point. 
[In  particular  a  straight  line  determines  oo J  lineal  elements,  all  hav 
ing  the  same  /-coordinate ;  while  a  single  point  (looked  upon  as  a 
line  curve  of  the  first  class)  determines  co  1  elements  all  having  the 
same  x-  and  jv-coordinates].  Such  a  single  infinity  of  lineal  elements 
is  said  to  form  a  union  of  clements,\  and  successive  elements  in  this 
case  are  said  to  be  united.  In  general  co  *  lineal  elements  do  not 
form  a  union  ;  it  is  easy,  however,  to  find  the  condition  that  they  do  : 

Two  relations  among  the  three  coordinates 

(113)  0(,r,  y,  /)  =  o  and  i//(.v,  y,  /)  =  o 

*  At  times  it  is  convenient  to  replace  the  line  by  its  direction  in  the  above  definition. 

t  We  shall  use  the  nonself-dualistic  set  (x,  y, /)  where  x  and  y  are  the  rectangular 
coordinates  of  the  point  and  /  is  the  slope  of  the  line. 

It  is  almost  needless  to  add  that  the  theory  here  developed  is  no  more  restricted  to 
this  choice  of  coordinates  than  the  general  theory  of  Analytic  Geometry  is  confined  to 
the  use  of  Cartesian  coordinates. 

J  In  this  case  the  locus  of  the  points  of  the  elements  coincides  with  the  envelope 
of  the  lines  of  the  elements ;  and  besides,  the  point  of  tangency  of  each  line  with  the 
envelope  is  the  point  of  the  element  to  which  the  line  belongs.  This  locus  will  be  re 
ferred  to  as  the  curve  of  the  union. 

175 


1/6  THEORY   OF   DIFFERENTIAL   EQUATIONS  §48 

determine  oo  1  elements.*     The  locus  of  the  points  of  the  latter 
(114)  o>O,/)=o 

is  obtained  by  eliminating/  between  the  two  relations  (113).  A 
union  exists  provided  the  value  of/,  in  terms  of  x  and  y,  obtained 
from  either  of  the  two  relations  is  the  same  as  that  of  the  slope  of 
the  tangent  to  the  curve  (114),  i.e. 

*"*~~~5' 

where  partial  differentiation  is  indicated  by  a  suffix.  The  condition 
that  the  lineal  elements  determined  by  (113)  form  a  union  is  therefore 
that 


(JI5)  dy 

Ex.  1.    Starting  with  the  relations 


the  point  locus  is  the  circle  x1  +  jy2  =  I.     Here 

£=-*=/. 

dx          y 
FIG.  6  Hence  the  elements  form  a  union.    (See  Fig.  6.) 

*  A  single  relation  (j>(x,  y)=o  free  of/  defines  ool  unions,  each  consisting  of 
the  oo  l  elements  having  a  point  of  the  curve  0  (x,  y)  =  o  in  common,  p  being  un 
determined. 

Hence,  if  neither  of  the  relations  <f>(x,  y)  =o  and  \f/(x,y)  =  o  involves  p,  they  to 
gether  determine  a  finite  number  of  unions,  each  consisting  of  the  oo »  elements  hav 
ing  in  common  a  point  of  intersection  of  the  curves  (f>(x,y)  =  o  and  \f/(x,y)  =  o.  (See 
Ex.  4,  below.) 

f  The  same  condition  obviously  holds  when  the  lineal  elements  are  determined 
parametically 

("6)  *  =  *(/),  y=  Y(f),p  =  p(t). 


CONTACT  TRANSFORMATIONS 


177 


Ex.  2.    In  the  case  of 

y  +  xp  =  o,   x  +  yp  =o 
the   point   locus   is   the   pair  of  lines   x2  —  y1  =  o. 

Here  -2-  =  -  ;   while  /  =  —  -  •     Hence  there  is  no 
dx      y  y 

union.     (See  Fig.  7.) 


Ex.  3.    In  the  case  of 

y  =  xp  -\-  i,   p  —  a  =  const. 

the    point    locus    is    the    line   y  —  ax  -|- 
—  a  —  p.     Hence  the  elements  form  a  union.     (See  Fig.  8.) 


FIG.  8 


Here 


dx 


Ex.  4.    In  the  case  of 


the  point  locus  is  the  point  x  =  o,  y  =  i,  while  /  is  undetermined.     The 
elements  form  a  union.     (See  Fig.  9.)  FIG.  9 

Ex.  5.   The  elements  determined  by 
x  =  cos  /,  y  =  sin  /,  /  =  tan  / 

do  not  form  a  union,  since   the  point  locus  is  the 
circle  x'2+y*=i,  where 

dx          y 

(See  Fig.  10.) 
FIG.  10 


Ex.  6.    In  the  case  of 


y- 


the  point  locus  is  the  line  y  —  I.    Along  this  p  =  O. 
Hence  the  elements  form  a  union.     (See  Fig.  n.) 


FIG.  ii 


THEORY   OF  DIFFERENTIAL   EQUATIONS  §§48,49 


Ex.  7.    In  the  case  of 

y  =  xp+l,  7  =  3 

the  point  locus  is  the  line  y  =  3.  Along  this  p  =  o. 
But  the  elements  along  this  line  determined  by  the 
first  relation  have  p  =  *- —  —  ^  o.  Hence  there  is 


I2  no  union.     (See  Fig.  12.) 


49.    Contact   Transformation.  —  Of    the    possible    transformations 
on  the  coordinates  of  a  lineal  element 

(117)        Xl  =  X(x,  y,  /),  y,  =  Y(x,  y,  /),  pl  =  P(x,  7,  /)  , 

those  which  transform  every  union  of  elements  into  a  union  play  an 
important  role  and  are  known  as  contact  transformations.  •  The 
condition  that  (117)  be  a  contact  transformation  is  readily  seen 
to  be 

(  1  1  8)        dy\  -  /!  dxi  =  p  (x,  y,  p)  (dy  —p  dx)  ,  where  p  =£  o. 

For,  from  the  condition  (115)  it  follows  that  if  a  union  is  to  be 
transformed  into  a  union  dy\  —  p\  dxv  must  vanish  whenever  dy  —  p  dx 
does  ;  that  is,  the  former  must  contain  the  latter  as  a  factor. 

Indicating   partial   differentiation    by  a    subscript,   (118)   may  be 
written 


This  is  equivalent  to 

(119)       Yf-PXt=-pp,    y,-JPX, 

whence 

(,20)  J'= 


§49  CONTACT  TRANSFORMATIONS  179 

and 

(121)  Xlt(Y,+pY,)-  Y,(Xf+pX,)  =  o* 

The  two  relations  (120)   and  (121)  may  be  put  in  the  compact 


V  V      i     j,    V 

Xp      X*  +P  Xy 

These  relations,  which  are  necessary  conditions  that  (117)   be  a 
contact  transformation,  are  also  sufficient,  as  may  be  seen  as  follows  : 
They  lead  at  once  to 


Y  —  PX       Y  —  PX 
or  *•*—  '  A*  =  £L  —  £21i  =  p.-j- 

-/  i 

Equations  (119)  follow  at  once,  and,  therefore,  condition  (118) 
is  fulfilled. 

Conditions  (120)  and  (121),  or  their  equivalents  (122),  may  thus 
be  used  instead  of  (118),  when  desired.}: 

*  Introducing  the  Poissonian  symbol 

r,n_   **  x,+px,\ 

=  yr  y.+tr,\- 

the  relation  (121)  takes  the  simple  form 
(121) 


When  two  functions  .Y  and  Y  satisfy  the  condition  (121),  they  are  said  to  be  in 
involution. 

f  This  value  of  the  common  ratio  p  cannot  be  identically  zero,  for  using  (122)  it 
may  be  written 

p  =  XpYy  —  XyYp  =  XxYy  —  XyYX  =   Xx  Y,,  -  A'r  Y.r  . 

XP  \r      pX,    '  pXp 


all  three  of  thr  numerators  cannot  vanish  simultaneously  since  .Y  and  Y  are  supposed 
to  be  independent  rum-lions. 

J  An  dement  transformation,  which  is  not  a  contort  transformation,  transforms  pre 
cisely  oo-  unions  into  unions.  (See  K;IMI<M,  American  Journal  of  idathematics, 
Vol.  XXXII,  p.  393).  Thus,  A"  =  .v,  Y  =  />,/'  v,  which  is  obviously  not  a  contact 
transformation,  transforms  the  union  defined  by  y  +  p  —  c^e*,  y—p  —  c.^-1  for  any 
pair  of  values  of  c±  and  c.±  into  a  union. 


180  THEORY   OF  DIFFERENTIAL   EQUATIONS  §49 

Remark  i.  —  Of  the  three  functions  X,  Y,  P  in  the  contact  trans- 

« 

formation  (i  1 7),  either  one  of  X  and  Y  may  be  selected  at  pleasure ; 
the  other  one  is  then  determined  as  a  solution  of  the  linear  partial 
differential  equation  (121).  With  X  and  Fselected,  P  is  determined 
uniquely  by  (122). 

The  extended  point  transformation  (§  13)  is  evidently  a  special 
case  of  a  contact  transformation.  For  if  X  and  Fare  any  functions 
free  of/,  (121)  holds;  while  the  form  for  the  accompanying  trans 
formation  ofjv'  or/,  given  by  (21),  is  exactly  (122). 

In  what  follows  we  shall  exclude  extended  point  transformations 
from  consideration,  unless  specific  mention  is  made  to  the  contrary. 

As  an  example  of  a  contact  transformation  may  be  mentioned  the  transforma 
tion  by  reciprocal  polars  with  respect  to  a  conic.  The  transformation,  in  case  the 
conic  is  the  circle  x~  +  y1  =  i,  takes  the  form 


Here  _ 

y(y  -  xp} 

The  transformation  by  reciprocal  polars  with  respect  to  the  parabola  x2  =  zy 
is  given  by 

(-#)  *\  —p-,  y\  =  xp  —  /,  p\  =  x. 

Here  dyi—p\dx\=—(dy  —  pdx}. 

In  the  above  illustrations  a  union  whose  curve  is  a  point  is  trans 
formed  into  one  whose  curve  is  a  straight  line.  That  in  the  case  of 
every  contact  transformation  (not  an  extended  point  transformation) 
a  union  whose  curve  is  a  point  \  must  be  transformed  into  one  whose 

*  These  equations  may  be  obtained  as  follows  :   The  point  (x,  y}  of  an  element 

(x,  y,  p)  is  transformed  into  the  polar  line  xx±  -\-yy\  =  i  whose  slope  is  p±  = •     The 

line  of  the  element,  ? 

Y-y=p(X-x)  or  ^2*-\ £— =  i, 

y-xp      y-xp 

is  transformed  into  the  pole 

~ y  —  xp'  yi     y  —  xp 
\  Excepting  possible  special  points ;  e.g.  the  origin  in  Ex.  3,  p.  185. 


§49  CONTACT  TRANSFORMATIONS  l8l 

curve  is  an  actual  curve  may  be  seen  by  eliminating  /  and  p\  from 
equations  (117).   There  results  from  this  elimination  a  single  relation,* 

(123)  F(x,y,x\,yl)  =  °, 

which  determines  a  locus  for  the  points  (#„  y^  corresponding  to  a 
fixed  point  (x,  y)."\ 

Moreover,  a  contact  transformation  is  determined  by  a  relation  of 
the  type  (123),  provided  the  three  equations 

(124)  F=o, 


can  be  solved  for  x,  y,  p,  and  also  for  xlf  y\,  JV     For,  solving  for 
#i»  y\i  P\»  there  results  the  transformation  of  the  three  variables 

(117)  x,  =  X(x,  yt  p),  y\  =  Y(x,  y,  /),  p,  =  -  ^. 

F* 

That  this  is  a  contact  transformation  may  be  seen  readily.     For 

from  F  F 

*—!?•*>  —  % 


*  If  there  were  two  independent  relations, 

Fl(*,y,  *i.^i)=  o,   F*(xty,  JT,,^)  =  o, 

they  could  be  solved  for  x\  and  y\  in  terms  of  x  and^y,  which  would  imply  that  (117) 
was  an  extended  point  transformation. 

f  We  may  say  (fixing  our  attention  on  the  curve  of  a  union)  that  the  effect  of  the 
contact  transformation  is  to  transform  any  point  (a,  l>)  into  the  curve  F(a,  b,  x^,y^)  =  o; 
while  a  point  transformation  transforms  a  point  into  a  point. 

Moreover,  it  is  not  difficult  to  show  that  a  contact  transformation,  in  general,  trans 
forms  a  union  determined  by  a  curve  C  into  one  whose  curve  O  is  the  envelope  of  the 
curves  into  which  it  transforms  the  various  points  of  C,  or,  using  the  same  form  of  ex 
pression  as  above,  we  shall  say  that  it  transforms  the  curve  C  into  O.  (Thus  sec  Lie, 
Berllhrungstransformationen,  p.  49)  .  If  it  should  happen  that  the  curve  C  is  one  of  the 
curves  F  (x,  y,  (t,  ft)  =  o,  where  Ot  and  /3  are  any  constants,  its  transform  C'  is  the 
point  («,  /3). 


1  82 


TIIKOKY   OF  DIFFERENTIAL   EQUATIONS 


§49 


Differentiating  (123)  gives 

FXi  dx±  +  FVi  d)\  =  —  (Fx  t/x  -f-  Fy  ffy) . 

p 
Hence  dy\—  p\dx±  =  —  -JL(ay—'paxyt 

f*i 

which  proves  that  (117)  is  a  contact  transformation. 

The  condition  that  (124)  be  solvable  for  x,  y,  /and  for  A-,,  y^  ft 
can  be  expressed  very  simply  analytically  : 

In  order  to  be  able  to  solve  for/  it  is  necessary  arid  sufficient  that 
Fy-=£o  when  F—  o.  Similarly,  F  =£  o  when  F  =  o  is  the  condition 
that  one  be  able  to  s%lve  for  ft. 

The  condition  that  the  first  two  equations  of  (124)  can  be  solved 
for  ^  and  }\  is  the  non-vanishing  of  the  functional  determinant 


or 


In  the  latter  the  factor  - -----  is  omitted  since  it  is  not  zero  whenever 

F=  o,  because  F  is  supposed  to  be  generally  analytic,  and  besides 
it  is  not  infinite  since  Fv=£o  when  F=  o,  by  hypothesis.  This  de 
terminant  can  be  put  in  the  more  symmetrical  form 


A  = 


Fx       Fy 


F      F       F 

!/i  xy.        *   y 


Since  A  contains  Fas  a.  factor  whenever  either  Fy  or  Fv  does,  the 
non-vanishing  of  A  when  F  —  o  assures  the  non-vanishing  of  Fu  and 
F  .  Hence  the  only  condition  that  (124}  be  solvable  for  x^  ylt  ft  is 


(125) 


o  when  F=o. 


Because  of  the  symmetry  of  A  as  to  x,  y  and  x\,y\,  (125)  is  also 
the  condition  that  (124)  be  solvable  for  x,y,p. 


§49 


CONTACT  TRANSFORMATIONS 


183 


Remark  2.  —  It  is  interesting  to  note  that  A  =£  o  is  the  condition 
that  F(x,  }',  .TJ.  _)',)  involve  x  and  y  as  two  essential  parameters,  and 
also  that  it  involve  xl  and  )\  in  the  same  way  ;  when  such  is  the 
case  F(a,  b,  xlt  }\)  =  o  defines  oo2  curves  for  all  choices  of  a  and  ^, 
and  F(xty,  a,  ft)  =  o  defines  oo2  curves  for  all  choices  of  a  and  /?. 
For  if  x  and  y  are  not  essential  parameters  in  F(xt  y,  x\,  y^,  two 
functions  of  x  and  y,  say  xi(x>  }')  an<^  Xz(x>y)>  can  be  found  such 
that  (see  Note  VII  of  the  Appendix) 


This  carries  with  it 


.•.  A  = 


o        F,       Fv 


\     FXVi     FUVi 
Conversely,  if  A  =  o 


f 


^     ~\~  \->F        F 


o. 


where  p  is  a  constant  as  far  as  xl  and  j^  are  concerned,  but  may  be  a 
function  of  x  and  y. 

Hence  A  =  o  carries  with  it  a  relation  of  the  type 

Fz  —  p  (x,  y)Fu  =  o, 

which  is  the  condition  that  .v  and  y  are  not  essential  parameters  in 
F(x,  y,  *„;'!). 

In  exactly  the  same  way  it  can  be  shown  that  if  A  =  o  xl  and  Vi  are 
not  essential  parameters  in  F. 


1 84 


THEORY   OF   DIFFERENTIAL   EQUATIONS 


i.,49 


The  equations  of  transformation  (<-/)  and  (B)  in  the  cases  of  transformation 
by  reciprocal  polars  given  above  are  readily  obtained  by  the  method  here  given 
when  jcx\  +  yy\  =  1  and  xx\_  —  y—  ji  =  o,  respectively,  are  selected  as  the 
relation  (123). 

For  the  transformation  by  reciprocal  polars  with  respect  to  the  general  conic 

ax*  +  2  hxy  +  by'1  +  2  gx  +  2fy  +  c  —  o 

the  relation  (123)  is  the  equation  of  the  straight  line 

fi(xyi  +  yxfi  +  %'i  +  g(x  +  *i)  +/(}'  +  yd  +  c  =  O, 
(ax  +  hy  +  g)xi  +(hx  +  by  +  f)y\  +  gx  +  fy  -\-c-O. 


Here 


AfEE 


o  ax\  +  Ay  i 

ax  -\-  hy  -f-  g  & 

hx  +  by  +f  h 


by\.  +  f 


Subtracting  ^ri-times  the  second  row  -f  >'i-times  the  third  row  from  the  first, 
and  taking  account  of  (123') 


gx  +fy  +  c  g  f 
a  x  +  hy  +  g  a  h 
hx  +  by  +/  h  b 


<  S  f 
g  a  h 
f  h  b 


a  h  g 
h  b  f 
S  f  ' 


i.e.  A  equals  the  discriminant  of  the  conic,  and  is  different  from  zero  in  case  the 
conic  is  an  actual  one  and  not  a  pair  of  straight  lines.  In  this  case  the  method 
given  above  applies.  Solving 


(ax  +  hy 
(a 


+(hx  +  by+f}yv  +  gx  +  fy  +  c  =  o, 

+  (h  +  bp}yi  +  g  +  fp  =  Q, 

ax  +  hy  +g+(hx  +  by  +/)/i  =  o, 


for  x\, 


i,  the  formulae  of  transformation  are 


_G(xp-y)+ff-Ap         _/-(xp- 

~  ~ 


_ 
p>  *         kx+ty+f 

where  A,  B,  C,  F,  (,\  //are  the  respective  cofactorsin  the  discriminant  of  the  conic. 
The  transformations  (.4)  and  (/?)  are  obviously  special  cases  of  (C). 


§§49,5°  CONTACT  TRANSFORMATIONS  185 

Another  interesting  contact  transformation  is  obtained  by  selecting  for  (123) 
the  equation  of  the  circle, 
(123")  O  -  xtf  +  (y-ytf  =  r*  ^  o. 

In  this  case  A  =—  8  r2,  and  the  equations  of  transformation  are  readily  found 
to  be 
(Z?)  *i  =  x  ±  —  &  —  .  yi  =y  T 


Vi-f/-  vi  +/1 

The  effect  of  (D)  is  to  transform  any  curve  into  a  pair  of  parallel  curves,  one 
on  each  side  of  the  original  one,  and  at  a  distance  r  from  it,  as  is  apparent  from 
the  nature  of  (123").  A  transformation  of  this  type  is  referred  to  as  a  dilatation. 

Find  the  contact  transformations  determined  by  the  following 
relations  : 

Ex.  1.    (x-xiy-2a(v-yl)=o.     E^    4    ^i+^i=I> 

Ex.  2.    ^^iT  +  IZ^i)!=I. 

Ex.  5.    *+*«!. 

•*!         Jl 

Ex.  3.    av'  +  Vf-Ov^  +  ri'O^o. 

50.  Group  of  Contact  Transformations.  Infinitesimal  Contact 
Transformation.  —  If  in  the  one-parameter  group 

(126)       xl  =  X(xtytpt  a),  y,=  Y(x,y,p,  a),  p,  =  P  (.v,  .v,  /,  a) 

the  condition 

(i  1  8)  <iY-PtiX  =  P(x,  y,  p)  (//r  -/./jf), 

or  its  equivalent  (122),  holds,  (126)  defines  a  ant-parameter  group  of 
c  'en  tact  transformations* 

Like  any  one-parameter  group  in  three  variabK-s  ($11)  the  group 

(126)  contains  an  infinitesimal  transformation 

(127)  .V!  =  A'  +  ^(A-,.V,/)  oa,  v,  =  v  +  iy  (.v,.v,/)  Sr/,/,  =/-f-7r(.v, 


1  86  THEORY   OF   DIFFERENTIAL   EQUATIONS  §50 

whose  symbol  may  be  written 

<'*>          ^€+'f^f- 

Thus  the  dilatations 


form  a  group,  with  the  infinitesimal  transformation 

£y-  p  df  _  I 

^T+J1  d*     Vi 
Similarly  the  transformations  (Ex.  I,  §  49) 

*i  =  *—  «/,  ;'!=/-  ^  , 

form  a  group,  with  the  infinitesimal  transformation 


Since  (127)  is  also  a  contact  transformation, 
(i  18')   d\\—pi  dx^—dy—p  dx+  (dy  —  p  <%  —  TT  <tx)$a*s=£p(dy—p<&). 

.-.  p  =  i  +  a-  (x,  y,  /)  &z, 
where 

(  I  29)   <r(x,  ;-,/)  (ity—p<tx)  =<h)—p<%—irt/x  =  ff(ii—p£)  -f^  t/p  —  irt/\. 

Writing  with  Lie  " 
(130)  r)-/>£=  -  /r(-v»  .'',/), 

where  /Fis  known  as  the  characteristic  function  of  the  infinitesimal 
contact  transformation,  the  identity  (129)  may  be  replaced  by 

//',  +  TT  =  o/,  -  ^Fv  =  IT,  -  /Fp  +  ^  =  o  ; 
whence,  making  use  of  (  1^0)  and  eliminating  <r, 


*  Here,  as  alw.ivs   it)   tin-  case  of  infinitesimal  transformations,  higher  [.ou.-rs  of  Sa 
are  nrtjlcctcrl. 


§5o  CONTACT  TRANSFORMATIONS  187 

Moreover,  for  all  choices  of  the  function  IV,  £,  rj,  TT  are  so  de 
termined  by  (131)  that  the  corresponding  infinitesimal  transforma- 


satisfies  the  condition  (118')  and  is  therefore  a  contact  transforma 
tion  ;  hence  the 

THEOREM.  —  Connected  with  every  infinitesimal  contact  transforma 
tion  t)ii'  re  is  a  cJiaracteristic  function  IV  =  —  77  +  /£,  in  terms  of 
which  the  transformation  is  given  by  means  of  (131).  Conversely, 
starting  with  any  function  IV,  the  relations  (131)  define  an  infinitesi 
mal  contact  transformation. 

In  terms  of  the  characteristic  function  the  infinitesimal  transforma 
tion  takes  the  form 

(132)      /i/=  \vt  ^+  (Pwr-  w)V-(  w, 

or  using  the  Poissonian  symbol  (§  49) 
(13*)  Bf 

Choosing  for  IV  the  form 


gives  the  infinitesimal  transformation 

Bj-      P       df 

Vi  +/5  ~d*     v 

which  belongs  to  the  group  of  dilatations  (/?). 
The  selection 


gives  the  infinitesimal  transformation 

**f—     7- -\ 


which  belongs  to  the  group 

*"=.,»=,-  ..,,/,  =  /.    CK*.».H9) 

/>- 


1  88  THEORY   OF   DIFFERENTIAL   EQUATIONS  §50 

When  W  is  linear  in  /,  the  corresponding  transformation  is  an 
extended  point  transformation.     For,  if 


[(24),  §  13] 

Another  fact  worthy  of  mention  in  connection  with  the  character 
istic  function  is  the  effect  upon  it  of  a  change  of  variables  when  the 
latter  is  effected  by  means  of  a  contact  transformation.  As  was 
noted  in  §  n,  the  introduction  of  the  new  variables 

[14']  x  =  F(xty,p),    V  =  *(xty,p),   P  =  *(x,y,p) 

causes  the  infinitesimal  transformation  (128)  to  take  the  form 


By  the    definition  o_f  the   charactertisic  function   (130)   its    form 
after  transformation  is 


-  yp). 


If  [14']  is  a  contact  transformation, 

dy  —  pdx=.  p(dy  —  p  dx)  , 

or  yx-pxx=-Pp,   yu-pxy  =  p,   yp-pxp  =  o; 

whence 

(133)         W(x,  y,  p)  =  p(tf-ri)  =  P(.v,  v,  /)  lV(x,  .v,  p). 

Of  course,  in  the  right-hand  member,  x.  v,  p  must  be  replaced  by 
their  values  in  terms  of  the  new  variables  given  by  [14']. 


§§50,51  CONTACT  TRANSFORMATIONS  189 

The  characteristic  function  for  the  group  of  dilatations  in  the  case  of  rectangu 
lar  coordinates  was  seen  to  be 

w 

Introducing  the  new  variables,  ' 


for  which  dy  —  p  dx  =  —  (dy  —  p  dx~), 

it  is  easy  to  verify  that 


W  =  -  VTT/  =  -  Vi  +  x*. 

On  the  other  hand  the  new  variables 


y  -  xp  y  —  xp  y 

for  which  dy—pdx  =  —  (ify  —  p  dx) 

y(xp  -  .i') 


cause  the  characteristic  function  to  assume  the  form 
W  =  -fi±£  =  (xp  -  y)  Vor*  +  j 


51.    Ordinary  Differential  Equations.  —  A  differential  equation  of 
the  first  order 
(i34)  f(x,y,p)  =o 

may  be  looked  upon  as  a  relation  among  the  three  coordinates  of  the 
lineal  elements  of  the  plane,  with  the  understanding,  however,  that 

(•35)  P  =  di- 

dx 

So  that  the  differential  equation  defines  oo2  lineal  elements  which 
[because  of  (135),  which  is  identical  with  (i  15),  >$  48]  are  arranged 
in  oo1  unions.  The  solutions  of  the  differential  equation  are  the 
equations  of  the  curves  of  the  unions. 

Since  all  the  lines  through  a  point  constitute  a  union,  in  which 
case  the  common  point  is  the  curve  of  the  union,  such  unions  must 
be  taken  into  account  when  looking  for  the  solutions  of  a  differential 


IQO  THEORY   OF   DIFFERENTIAL   EQUATIONS  §51 

equation.     Thus  if  the  relation  (134)  is  free  of/,  say 


this  may  still  be  looked  upon  as  a  differential  equation  in  which  /  is 
arbitrary.     Such   a   differential  equation  defines,  besides   the   union 

whose  curve  is /"(.*,  y)  =  o,  those  unions 
determined  by  each  of  the  various  points 

•  ^V--\     of  the  curve.     See  Fig.   13.     Each  of 

these  points  will  be  considered   as  an 
p  integral  curve  of  the  differential  equa 

tion. 

Since  every  lineal  element  of  the  envelope  of  a  family  of  <x> l  curves 
is  an  element  of  some  curve  of  the  family  (compare  EL  Dif.  Kq. 
$$  29,  30),  the  equation  of  the  envelope  must  also  be  a  solution 
(i.e.  the  singular  solution)  of  the  differential  equation  of  the  family  of 
curves.  In  the  special  case  of  a  differential  equation  of  the  type 
(134')  the  curve  f(x,  y)  =  o  maybe  looked  upon  as  the  envelope, 
and  its  equation  is  therefore  the  singular  solution. 
The  Clairaut  equation  (El.  Dif.  Rq.  §  27) 

when  transformed  by 

(/?)  jf!  —p,  y\  =  xp  —  y,  f>i  =  x, 

takes  the  form  y\  +/(•* "0  =  °» 

which  is  of  the  type  (134').  It  has  for  integral  curves  the  various  points  of  the 
curve  vi  -f /(•*"!)  =  o,  while  the  equation  of  this  curve  itself  is  a  singular  solution. 
Passing  hack  to  the  original  variahles,  this  curve  is  transformed  into  some  curve 
<()(jc,y}  =  o,  and  its  points  are  transformed  into  the  tangents  of  0f.r,  y)=O. 
Their  respective  equations  are  the  singular  and  particular  (in  the  aggregate, 
general)  solutions  of  the  original  differential  equation. 
The  special  Clairaut  equation 

y  —  xp  —  rVi  +  p1  =  O, 

when  transformed  hy  the  dilatation 


takes  the  form  y\  —  x\p\  —  o. 


§§5  1,  5  2  CONTACT  TRANSFORMATIONS  19  1 

This  simple  differential  equation  has  the  obvious  general  solution  y\  =  cx\, 
which  is  the  equation  of  the  family  of  straight  lines  through  the  origin.  The 
envelope  of  this  family  is  the  origin,  which  determines  a  union  that  is  obviously 
consistent  with  the  relation  defined  by  the  differential  equation.  Passing  back 
to  the  original  variables,  the  origin  goes  into  the  circle  x-  +/-  =  r'2,  which  equa 
tion  is  therefore  the  singular  solution'of  the  original  differential  equation,  while 
the  lines  through  the  origin  go  into  the  tangents  to  this  circle.  The  equation 
of  their  family,^  =  ex  —  rVi  -f  c*  =  o,  is  the  general  solution. 

52.  First  or  Intermediary  Integrals.  —  The  differential  equation 
(136)  4>(x,y,p)  =  a, 

for  each  value  of  the  arbitrary  constant  #,  has  oo1  integral  curves. 
Allowing  a  to  take  successively  all  possible  values,  (136)  determines 
oo  "  curves  which  are  the  integral  curves  of  the  differential  equation 
of  the  second  order 

(•37)  g  =  *.  +  ^  +  *,J=<>. 

The  differential  equation  of  the  first  order  (136)  is  known  as  a  first 
or  intermediary  integral  of  (137).  From  the  above  it  is  seen  that  a 
first  integral  of  a  differential  equation  of  the  second  order  classifies  the 
oo  -  integral  CI/ITCS  of  the  latter  into  ^families  of  oo1  curves  each. 

This  classification  is  different,  of  course,  for  different  first  integrals, 
of  which  there  is  an  indefinite  number.  For 

(138)  *(x,y,p)=t 

will  also  be  a  first  integral  of  (137),  if,  and  only  if, 


is  the  same  as  (137),  i.e.  provided 


or 


(121) 


'p       *X+lJW» 


0. 


1  92  THEORY    OF    DIFFERENTIAL    EQUATIONS  §52 

Hence 

THEOREM  I.      The  necessary  and  sufficient  condition  tJiat 


be  first  integrals  of  the  same   differential  equation  of  the  second  order 
is  that  <f>  and  \j/  be  in  involution  (§  49). 

Starting  with  the  function  <£(.v,  _v,/),  a  second  function  \j/(x,  y,  p) 
will  be  in  involution  with  it  provided  it  satisfies  the  linear  partial  dif 
ferential  equation 

(139) 


This  linear  equation  in  three  independent  variables  has  two  inde 
pendent  solutions,  one  of  which  is  <t>(xtyyp).  All  of  its  solutions  are 
functions  of  these  two.  Hence 

THEOREM  II.  Knowing  <£(~r,  v,  /)  =  a,  a  first  integral  of  a  differ 
ential  equation  of  the  second  order,  all  of  its  first  integrals  may  be 
obtained  by  solving  the  linear  equation  (139).  Having  found  a  solu 
tion  of  (139),  independent  of  <j>,  all  the  first  integrals  are  given  by 


where  <J>  is  an  arbitrarily  chosen  function  of  $  and  ty. 
Since  two  independent  first  integrals 

<£(.v,  y,  p)=a  and  i//(.v,  v,  /)  =  b 

of  a  differential  equation  of  the  second  order  define  the  same  set  of 
co2  integral  curves  but  classified  in  distinct  manners,  for  a  particular 
but  arbitrary  choice  of  a  and  />,  say  an  and  /v  the  differential  equations 

(140)  <£(.¥,  y,  /)  =  tf(,  and  i/>(.v,  v,  /)  =  t>Q 

will,  in  general,  have  an  integral  curve  in  common.  At  each  point  of 
this  curve  both  equations  (140)  determine  the  same  value  of/; 


§52  CONTACT  TRANSFORMATIONS  193 

hence  the  equation  of  the  curve 

<o  (x,  y,  00,  /;())  =  o 

may  be  obtained  by  eliminating/  from  (140). 

Still  keeping  tf0  fixed  but  allowing  b  to  be  an  arbitrary  constant, 
the  result  of  eliminating/  from  <f>  =  #0  and  \\i  =  b  gives 

CD  (x,  y,  a0,  b)  =  o, 

a  solution  of  <£(jc,  y,  /)  =  #0  containing  an  arbitrary  constant  which 
is,  therefore,  its  general  solution.     Hence, 

THEOREM  III.     If  a  second  differential  equation 
*<*JVJP>.-.<* 

involving  an  arbitrary  constant  can  be  found  sucli  tJiat  ff>  and  \\i  are 
in  involution,  the  general  solution  of 

4>(x,y,P)  =  a* 

can  be  found  by  eliminating  />  from  the  ftvo  differential  equations. 

This  process  is  frequently  of  service.      (See  El.  Dif.  Eq.  §§  25,  26). 
Eliminating/  from  (136)  and  (138)  gives 

w  (x,  y,  a,  b}  =  o, 

a  solution  of  (137)  involving  two  arbitrary  constants.     It  is  there 
fore  the  general  solution.     Hence, 

THEOREM  IV.  If  two  independent  first  integrals  of  a  differential 
equation  of  t)ie  second  order  can  l>c  found,  its  general  solution  is  ob 
tained  I>\  eliminating  p  from  tJic  ecj  nations  of  the  first  integrals. 

Remark,  —  If  <£  and  \j/  are  in  involution,  it  follows  at  once  from 
the  above  that  the  two  relations 


determine  an  element  union  (§  48)  for  all  choices  of  the  constants 
a  and  b. 


194  THEORY   OF  DIFFERENTIAL   EQUATIONS  §§52,53 

It  should  be  noted,  however,  that 
(113)  0(^',  v,/)  =  o,  *t,(x,y,p)  =  o 

may  determine  a  union  without  the  identical  vanishing  of  [^>,  \j/~\  • 
thus  see  Ex.  6,  §  48.  But  in  every  case  when  the  relations  (113)  de 
termine  a  union,  [^,  <£]  must  equal  zero,  either  identically  or  because 
of  these  relations.  This  follows  readily  from  the  fact  that  whenever 
(113)  determine  a  union,  the  equation  of  the  curve  of  the  latter  is  an 
integral  curve  common  to  the  two  differential  equations  <£  =  o,  ^—  o; 
and  conversely. 

53.  Differential  Equation  of  the  First  Order  Invariant  under  a 
Group  of  Contact  Transformations.  The  general  type  of  differential 
equation  of  the  first  order  invariant  under  the  group  whose  infinitesi 
mal  transformation  is 


is  obtained  (compare  §  18)  by  equating  to  zero  the  general  solution* 
of  the  linear  partial  differential  equation 


On  the  other  hand,  the  condition  that  the  differential  equation 
/(•v>  r5  /)  =  °  De  invariant  under  the  group  whose  infinitesimal 
transformation  is  Bfis  obviously  ([12],  §  n) 

Bf=  o  whenever  f=  o. 

As  was  noted  in  §  51,  a  differential  equation  of  the  first  order 
(142)  /=<o(.v,y) 

arranges  the  oc2  lineal  elements  determined  by  it  in  oc1  unions,  the 
curves  of  which  are  its  integral  curves.     If  (142)  is  left  unaltered  by 

*This  solution  is  obviously  the  general  expression  for  \\\e  first  differential  invariants 
of  the  group,  the  name  given  by  Lie  to  invariant  functions  of  x,  y,  p. 


§53  CONTACT  TRANSFORMATIONS  195 

a  contact  transformation 

(117)        #,  =  X(x,  r, /),  yl  =  Y(x,  v,  />),  /,  =  7>(.v,  r,  /), 

the  latter  interchanges  the    integral  curves  of  (142)   among  them 
selves,  since  it  transforms  unions  into  unions. 

As  far  as  the  differential  equation  (142)  is  concerned,  the  only 
lineal  elements  operated  upon  by  (117)  are  those  whose  coordi 
nates  are  (.v,  r,  /  =  o>(.v,  r)).  These  elements  are  transformed  into 
(.v,,  Vi,  /,  —  o)(.v,,  v,))  by  (117),  since  the  latter  leaves  the  differ 
ential  equation  unaltered.  Hence  the  effect  of  the  contact  trans 
formation  (117)  on  the  differential  equation  is  the  same  as  that  of 
the  point  transformation 

(143)  #!  =  X(x,  y,  w(x,  y}),  y\  =  Y(x,  y,  «(#,  y)\ 

Whence  the 

THEORKM.  —  If  the  differential  equation 

(142)  f>=o>(x,y) 

is  invariant  under  tlie  contact  transformation 

(117)  a-,  =  A'(*,  y,  /),  yi  =  y  (x,  y,  /),  pl  =  P(x,  y,  /), 

it  is  also  invariant  under  the  point  transformation 

(143)  x}  =  X(x,  y,  o>(.r,  v)),  y\  =  Y(x,  y,  o>(.v,  y)). 
Both  transformations  interchange  the  integral  curves  <•>/"(  142). 

It  follows  at  once  that  if  the  differential  equation 
p  =  w(#,  y) 

is  invariant  under  a  group  of  contact  transformations  whose  infinitesi 
mal  transformation  is 

Bf=  «.v,  r.  / 1  V  •  |  • ,,,  .v.  y,  fy  %  +  ,(.,;  y,  /)  % 


196  THEORY   OF   DIFFERENTIAL   EQUATIONS  §53 

//  is  also  invariant  under  the  group  of  point  transformations  whose 
infinitesimal  transformation  is 

Uf=  t(x, y,  <r,  v)) f£ + ?(*, y,  «(*, y))*j-y- 

Either  of  the  methods  of  §§  12  and  20,  Chapter  II,  may  then  be  em 
ployed  for  solving  the  differential  equation. 

Remark.  —  Since  BW=—  WyW,  it    follows    that  the  differential 
equation 
(144)  W(xty,f)  =  <> 

is  invariant  under  the  group  of  contact  transformations  whose  infini 
tesimal  transformation  has  W^for  characteristic  function. 

But  the  invariance  is  of  a  special  kind.  The  effect  of  this  infinitesi 
mal  transformation  is  to  carry  the  point  (x,  )')  of  an  element  (x,  y,  p) 
into  (x  +  £  8a-j  y  +  77  &z)  where  $  =  IVP,  TJ  =plVp  —  W.  The  slope 
of  the  line  joining  these  points  is 

f -/--£=/ when  UK- <k 

I  wp 

Hence  any  element  whose  coordinates  satisfy  (144)  has  its  point 
carried  in  the  direction  of  the  line  of  the  element,  that  is,  the  ele 
ment  and  the  one  into  which  it  is  transformed  are  united  (§  48). 
The  infinitesimal  transformation,  therefore,  leaves  unaltered  each  of 
the  unions  (§  51)  determined  by  the  differential  equation  (144),  and 
the  group  has  this  effect  on  each  of  the  integral  curves  of  (144). 
Such  a  group  is  said  to  be  trivial  with  respect  to  the  differential 
equation  (144),  (§  12),  an^i  is  of  no  service  in  solving  it. 


APPENDIX 

NOTE   I 
THE   INFINITESIMAL  TRANSFORMATION 

In   case   both  —  (j>(x,  y,  a)  and  —  $(x,  }',  a)    vanish    identically 
da  da 

for  the  special  value  of  a  =  a0,  or  if  either  of  them  becomes  infinite 
for  that  value  of  a,  irrespective  of  the  values  of  x  and  y  that  may 
enter,  a  modification  of  the  process  for  finding  the  infinitesimal  trans 
formation  employed  in  §  2  must  be  made.  It  should  be  noted  that 
they  cannot  both  vanish  identically  for  all  values.  of  a,  for  in  that 
case  neither  of  the  functions  <£  and  if/  could  involve  a  at  all  ;  nor  can 
either  one  of  them  become  infinite  for  all  values  of  x,  y,  ami  a,  since 
<f>  and  i//  are  supposed  to  be  generally  analytic,  which  implies  the 
existence  of  finite  derivatives,  except  perhaps  for  special  values  of 
the  arguments. 

Let  a  be  a  value  of  the  parameter  for  which  —  and  -*  are  finite 

da  da 

and  at  least  one  of  them  different  from  zero.  The  transformation 
Ta  determined  by  it  has  for  inverse  a  definite  transformation,  7^,  of 
the  group,  corresponding  to  the  value  «  of  the  parameter,  where  a  is 
a  function  of  a  only.  Since  TzTa  =  Tn  is  the  identical  transforma 
tion,  TaTa+da  is  an  infinitesimal  transformation.  If  T*  is 


*\  =  4>(x,y,  «)i  J'i  =  $(*,  y,  «), 

the  infinitesimal  transformation   7i7*a+sa  may  be  written,  when  ex- 

197 


198  THEORY   OF   DIFFERENTIAL   EQUATIONS 

panded  by  Taylor's  Theorem 

x,  =  <f>(xlt  ylt  a  +  Sa)  =  x  +       <j>(xlt  }\,  u)8a*  -\  ---- 

j,  =  >/'(>„  }\,  a  -f  8tf)  =  >•  +  —  ^(xlf  }\,  a)8a  -f  •••, 

since  <£(.TI,  ji,  «)  =  ^,  ^(xltylt  a)—y.  Owing  to  the  way  in  which 
a  was  chosen,  neither  of  the  coefficients  of  Ba  is  infinite  for  all  values 
of  x  and  y,  and  one  of  them,  at  least,  is  not  identically  zero. 
Writing 


',  a)  ,  «]  =  ^(.r,  ;',  a)  , 
,J'?«),  «]^>;Gv,j,  a), 


(I45) 


it  follows  that  ^//  infinitesimal  transformation  of  the  group  (1}  of  the 

type  (2),  §  2, 

(2)  £v  =  £8rt,   8>'  =  17  80 

r^«  always  be  found.  • 

The  forms  for  ^  and  77  found  in  §  2  are  exactly  what  the  above 
become  fcfr  the  special  choice  a  =  a  =  0,,. 

From  the  above  it  is  seen  that  £  and  rj  in  (2)  depend  upon  the 
choice  of  «.  It  remains  to  show  how  they  depend  upon*  the  choice 
of  the  parameter.  Let 

8*  =  HO,  j)&7,    8r  =  H( 


or  xl  =  x+  s,(x,  .v)80,  ji',  =  y  4-  H  (.v,  j)8<7 

be  some  known  infinitesimal  transformation  of  the  group  (i),  where 
E  and  II  are  not  both  identically  zero,  and  neither  of  them  is  infinite, 
in  general.  The  result  of  performing  successively  any  transformation 
Ta  of  the  group  (i)  and  the  above  infinitesimal  transformation  is 


Hcru    J     0  (Vj.^'j,  (/.)  stands  for 


da 


,  a) 


AIT  KN  I.)  IX  199 

some  transformation  of  the  group  whose  effect  on  the  variables  x 
and  y  differs  from  that  of  Ta  by  an  infinitesimal  amount.  In  other 
words,  it  is  a  transformation  Ta+±a,  where  Art  is  an  infinitesimal  which 
is  a  function  of  a  and  Srt  only,  because  of  the  group  property  of  (i). 
From  the  first  definition  of  this  transformation 

#2  =  xi  +  »(!,  }\)&a  =  <£O,  )',  «   4-  H(  < 


while  from*the  second  definition 
x.z=<j>(x,yt  rt  +  Art)  = 


'   da 
Hence 


da 

A,t, 
Art 


«(</>,  i/OStf  A* 


(146) 


for  all  values  of  :r,  i1,  rt  and  Srt,  Art  being  a  definite  function  of  a  and 
Srt,  and  an  infinitesimal  along  with  Srt.  By  hypothesis  H  and  H  do 
not  both  vanish  identically;  suppose,  to  fix  the  ideas,  that  E  =£  o. 
It  follows  that  A*  is  not  left  unaltered  by  all  the  transformations  of 

the  group  (  r  )  ;  hence  <f>  must  involve  rt,  and  -*  ^  o.     With  a  proper 

aa 
choice  of  .v,  v,  rt  the  coefficient  of  8a  and  that  of  Art  in  at  least  the 

first  of  the  two  relations  (  i.j(>)  are  different  from  xero.  ]\y  a  theo 
rem  in  the  Theory  of  l-'unctions,  concerning  the  inversion  of  power 
series,  Art  is  developable  in  powers  of  Srt,  the  development  beginning 
with  the  first  power.  Hence 


r,  A,/  is  ri  (unction  of  ,1  ;intl  8a  only,  the  rorfticirnts  in 
',v  iind  an-  tree  ot  r  .ind  v. 


200  THEORY  OF   DIFFERENTIAL   EQUATIONS 

where  w(a)^o.  A<z  is  thus  of  the  same  order  of  infinitesimals  as 
8a.  Putting  this  value  in  (146),  dividing  by  8a,  and  passing  to  the 
limit  &a  =  o, 

(147)  E(*,*)=7*(«) 


or  remembering  that     x  =  <b(x^  jr1}  a),  y  =  if/(xlt  )\,  a), 
these  may  be  written 


Using  (145),  and  replacing  ^  and  j',  in  these  identities  by  x  and  y, 

we  have 

(148)     £(*,^,0)  =  _iE(*,j'),    ^0,,fl)  =  _l_H(^^). 


The  effect,  then,  of  using  different  values  of  the  parameter  in  deter 
mining  an  infinitesimal  transformation  by  the  method  of  the  first 
part  of  this  note  is  to  obtain  pairs  of  coefficients  of  8a  in  the  two 
formulae  which  are  proportional,  the  factors  of  proportionality  being 
constants.  Hence,  by  Remark  i,  §  2,  all  the  infinitesimal  transfor 
mations  so  obtained  are  one  and  the  same.  We  have  thus  arrived 
at  the 

THEOREM.  —  Every  one-parameter  group  of  transformations 
i 
x\  =  4>(x,  y,  a),    Vi  =  ^O,  }',  a) 

has  one  and  only  one  independent  infinitesimal  transformation 

&x  =  £  (  xt  y  )  8tf  ,   By  =  rj(  -v,  v  )  Sa, 
where 

.v  v  «        » 

da 


APPENDIX  201 

and  <t  is  any  value  of  the  parameter  such  that  at  least  one  of  f  ^  \   and 

[-JL]    is  not  identically  zero,  and  neither  of  them  is  infinite  for  all 
\daja 

values  of  x  and  y. 

In  general  #0  is  a  possible  value.  In  §  4  is  shown  that  the  trans 
formations  of  the  group  can  always  be  put  in  such  form  that  this  is 
true.  When  for  a  given  group  this  value  cannot  be  used,  this  is  due 
to  the  way  in  which  the  parameter  enters,  and  is  not  a  peculiarity  of 
the  group. 

Remark.  —  This  theorem  and  its  proof  hold  for  n  variables  without 
any  but  obvious  modifications  to  take  account  of  the  number  of 
variables. 

NOTE    II 
SOLUTION  OF  THE  RICCATI  EQUATION 

z^ 
air 

In  §  1  8  the  general  method  for  finding  the  differential  equations 
invariant  under  a  given  group  led  to  the  solution  of  the  Riccati 
equation 

(39) 


dx      tdx     t\df      dx  id 

in  which  y,  wherever  it  occurs,  is  supposed  to  have  been  replaced  by 
its  value  in  terms  of  x  and  c  [say  y  =  <£(.v,  '")]  obtained  from 
u(x,  y)  =  c,  the  solution  of  the  differential  equation 


It  is  very  easily  seen  that 
(149)  y 


202  THEORY   OF   DIFFERENTIAL   EQUATIONS 

in  which  v  is  replaced,  as  above,  by  <£(.v,  r),  is  a  particular  solution 
ot   (39).     For  differentiating  (149) 


dx       £  \dx       dy  dx)       ^  \dx      dy  dx) 
Remembering  that  ^-  =  3,  this  becomes 

dx       k 


Whence  follows  at  once  that  (149)  satisfies  (39). 

It  is  a  well-known  fact  that  the  knowledge  of  a  particular  solution 
of  a  Riccati  equation  enables  one  to  fiml  a  transformation  of  variables 
which  reduces  the  equation  to  a  linear  differential  equation  of  the 
first  order,  whose  solution  requires  two  quadratures  (see  EL  Dif.  Eq. 
§  73,  i°).  For  the  sake  of  simplicity,  writing  (39)  in  the  form 


and  its  particular  solution       •  y'  =y0't 
the  transformation  /  =  -  +  yd 


changes  the  differential  equation  into 
£  +  (Xl  -f  2  yJXJz  +  X,  =  o, 

which  is  linear.     If  z—  w(.r,  k)  is  the  solution  of  this  equation, 

(150)  /  =  __1_+>?1^(^)J 

*(x,k)  ^ t[.x, +(x,  c)] 

is  the  solution  of  (39).     Solving  (150)  for  k,  and  replacing  <£(.r,  c) 

by->''  «'(x,y,S)=* 

is  the  required  second  solution  of  (37),  §  18. 


AITKXDIX  203 


^=x*=  5 

—  y       x        I  +y* 


tf.,i  j  _|_  v- 

The  Kiccati  equation  (39)  is  -  •  .  -  =  -• 

ax      _  vV  — 


The  transformation 

s        _v 

dz    .      -zx  I 

reduces  this  to  •  -  2  =  -  — 

^r      c  —  x~          Vf  —  x- 


—       "     .      7  / 

Integrating,  -  +  k{c  — 


I  lence  /  =         .   ^T^T^T  ~  ~ ' 

and  k  =  (TT^TT^  -u  ^  ^  W'  ^' 7'  ^'^ ' 

Compare  this  with  II,  §  19. 


NOTE   III 

ISOTHERMAL   CURVES 

The  condition  that  two  distinct  families  of  curves 
<£(.v,  j)  =  const,  and  i^(,v,  y)  =  rrv/y/. 

divide  the  plane  into  infinitesimal  squares  may  be  obtained  from  the 
following  considerations  :  * 

Passing  to  the  new  system  of  coordinates 


the  two  families  of  curves  have  the  simple  equations 
x  =  cons/,  and  y  =  const, 

*AH  this  holds,  practically  without  change,  for  isothermal  curves  on  surfaces. 


204  THEORY   OF   DIFFERENTIAL   EQUATIONS 

From         dx  =  ^  dx  +  3*  dy  and  dy  -  *  dx  +  d*  dy, 
dx  dy    "  ox  dy 


dy  dy  dx  dx 


A+> 

-here 


the  Jacobian  of  <£  and  ^,  which  is  not  identically  zero,  since  the  two 
families  of  curves  are  distinct.  (See  El^  Dif.  Eq.  Note  I  of  the 
Appendix.) 

The  expression  for  the  element  of  length  of  arc  of  any  curve  in  the 
plane,  in  terms  of  the  new  coordinates,  is 

d?  =  do?  +  <if=  Edx*-2Fdx  dy  +  G  dft 
where  the  coefficients 


dy  dxdxdydy     G  =  \dxj       \dy 


U  ($> 

are  to  be  expressed  in  terms  of  x  and  y  by  aid  of  (151). 

A  first  requirement,  that  the  two  families  of  curves  form  isothermal 
systems,  is  that  they  cut  each  other  orthogonally.     The  condition  for 

this  is  1,1, 

d<£      d\l/ 

dx      dy 

-T4>  =  ty'orF=°- 

dy      dx 

Hence  a  necessary  condition  is  that  the  expression  for  the  element 
of  length  of  arc  assume  the  form 

(/s-  =  E  dx1  + 


APPENDIX  205 

For  a  curve  of  the  family  x  =  const,  (which  will  be  referred  to  as  an 

jr-curve) 

dsx  =  ^J  G  dy, 

while  for  a  y-curve  ds   =  "y£  dx. 


If  -\/~E  =  V£  at  every  point  in  the  plane,  the  curves  divide  the 
plane  into  infinitesimal  squares,  for  choosing  dx  the  same  as  dy, 


Moreover,  if  V  '  E  and  V  G  contain  a  common  factor,  and  each  of 
the  remaining  factors  is  a  function  of  the  corresponding  variable 
only,  thus 


AOr,  »)/?(,), 

the  introduction  of  the  new  variables 

X=Ca(x)itx  and   Y=  Cft(y)tfy 
gives  </jT=A(Ar,  Y)JXandJsT  =  A(X,  Y)dY9 


where  A(X,  Y)  is  what  \(x,  y)  becomes  when  x  and  y  are  replaced 
by  their  values  in  terms  of  X  and  K     The  families 

X=  const,  and  Y=  const. 

(which  are  obviously  the  same  as  x=  const.  and  y  —  const)  have  the 
desired  property.     Hence  the 

THEOREM.  —  The  necessary  and  sufficient  condition  that  tlie  entires 
<f>(x,  _)')  =  const,  and  thrir  orthogonal  trajectories  i//(.v,  \}—  const. 
divide  the  plane  into  infinite  si  inal  squares  is  tJiat  tJie  choice  of  variables 


reduces  the  expression  for  the  element  of  length  of  arc  to  the  form 

,K-  =  X2(jr,  y}  ]  [«(jr)//r]2  +  [0(y  ^]2f  , 
where,  in  particular,  <c(x)  and  J3(y}  tnav  eacli  he  unity. 


2O6  THEORY   OF    DIFFERENTIAL    EQUATIONS 

Thus  in  the  case  of  a  family  of  concentric  circles  and  their  orthogonal  trajec 
tories, 

x  =  x2+y*,  y--, 


2x 

Putting  X  =  log  Vjr,    Y  —  tan'1!/, 


For  other  examples  of  isothermal  systems,  see  §  24. 

NOTE    IV 

DIFFERENTIAL   EQUATION   OF   THE   SECOND   ORDER  NOT 
INVARIANT   UNDER  ANY  GROUP 

If  the  differential  equation  be  written  in  the  form 
the  condition  that  it  be  left  unaltered  by  the  group 

(  6 1 ' )      —  £ rj  -  —  T/— -  + 17"  =  o,  whenever  /'  =  F  (x,  y,  /  ) , 


t(5S),ia63, 


~- 


Al't'KMtlX 


207 


Replacing  y"  t  wherever  it  occurs  in  y",  by  F(x,y,  y')  the  condi 
tion  (61')  becomes 


fr?_  _         _          -. 

dy     2  ** 


- 
dx'By 


ty          W 


-v 


dy 


for  all  values  of  x,  y,  y'  . 

Since  (152)  is  an  identity  with  respect  to  x,  y,  and  y',  it  is  equiva 
lent  to  a  number  of  differential  and  finite  equations  in  £  and  ry,  the 
exact  numl^er  depending  on  the  form  of  F.  Fixing  one's  attention 
ony  alone,  (152)  is  equivalent  to  at  least  four  equations,  and  per 
haps  more.  In  general  it  is  impossible  to  find  functions  £(x,  y)  and 
ij(x>  }')  to  satisfy  all  these  conditions. 

As  an  example,  consider  the  differential  equation 


The  identity  (152)  leads  to 


tan/. 


dy 


=  0, 


the  dots  standing  for  terms  free  of  tan  y'  and  sec  y  and  involving 
second  derivatives  of  £  and  tj.  (See  below.)  This  identity  implies 
the  following  relations  : 


208  THEORY  OF   DIFFERENTIAL   EQUATIONS 


«!=<>• 


From  (#)  and  (/)  -~  =  3-  -  =  o. 


These  together  with  (V)  and  (</)  make  it  necessary  that 
£  =  const,  and  7;  =  <r0«j/. 

Hence,  the  omission  of  terms  involving  higher  derivatives  of  £  and 
rj  above. 

Since  (6)  must  hold  for  all  values  of  x  and  y 

£  =  r)  =  o; 

i.e.  there  is  no  infinitesimal  transformation  and,  therefore,  no  group 
that  leaves  the  differential  equation  unaltered. 

Remark.  —  The  case  of  a  differential  equation  of  the  first  order  is 
entirely  different.     The  condition  that 


be  invariant  under  6/=  £  ,.    +  77  J- 

ox      '  oy 

may  be  put  into  the  form 


Here  one  of  the  functions,  say  £,  may  be  chosen  at  random,  leav 
ing  a  partial  differential  equation  in  T;,  which  always  has  a  solution  ;  as 
a  matter  of  fact  it  has  an  indefinite  number  of  them.  This  is  in 
entire  accord  with  the  result  arrived  at  in  §  15. 


APPENDIX  209 

NOTE   V 


The  symbol  of  the  infinitesimal  transformation  of  the   extended 
group  corresponding  to  6^=  £  7)    ~^~  ^  ;}    ^s  C(24)>  §  T3] 


dx         By\dx          dy 
Introducing  the  symbol 


£//"may  be  written  in  the  form 


and  (BU')f  =  J}(Jl/+  P(X,  y,  /)       , 

where  p(x,y,y')=  B(By  —yfJB£)  is  some  function  of  x,y,  y',  whose 
actual  form  is  of  no  importance  in  this  discussion.  Introducing  the 
additional  symbol 


be  written  in  the  form 


Also  (CU)f=<r(xtyt]f)Cf9 


where  cr(.v,  v,  r')  =  C(fty  —  y'B£)  is  also  a  function  of  .v,  r,  r',  whose 
form  is  of  no  importance  here.  The  fact  to  be  emphasized  is  that 
{Blf)f  and  (  Ct?}/  arc  linear  functions  of  Bf  and  Cf,  the  coefficients 
being  functions  of  x,  y,  y'  . 


210  THEORY   OF   DIFFERENTIAL   EQUATIONS 

Moreover,  if  U^f  and  U.>f  are  any  two  groups,  (/?(£//£/,'))/  and 
(C(C/i  &>))/  are  also  linear  in  Bf  and  Cf.  For  from  Jacobi's 
identity  (§  36) 

(B(  U{  U{)  )/+  (  U{  (  MB)  )/+  (  Ui  (BUI)  )/  =  o, 


and  in  an  analogous  manner, 

(  C(  U{  W)  )/  =  (  U^,  -  U^  Cf. 

Since  (U^U.^f  is  of  the  same  type  as  Uf  (§  14),  it  may  be  written 


and 


Noting    that    (U{  U*)f  coincides   with    (U^)'/  in    the    first   two 
terms,  at  least,  we  may  put 


It  remains  to  show  that    o>  =  Brj 
The  alternant  of  /?/and  ((SiC7»)/is 


This  being  linear  in  /y/and  Cf,  as  was  proved  above, 


.'.  Bt  =  A,     />'ry  -  <u  =  A/. 


.\ITK\ni\  211 

Whence  co  =  ^-/^. 

This,  as  was  noted  above,  establishes  the  identity 


Remark.  —  It  can  also  be  proved  that  for  the  ;«-times  extended 
groups  ^"" 


NOTE   VI 

CONTINUOUS  GROUPS   INVOLVING  MORE  THAN  ONE 
PARAMETER 

r-parameter  Group  of  Transformations.  —  The  aggregate  of  all  the 
transformations  * 

x1=  4>(x,y,  0i,  tfjj,  •••,  ar), 

(153) 

h'l  =  t('X>}'>   al>   a*    •",   <*r)> 

obtained  by  assigning  to  the  parameters  alt  a.2,  •••,  aT  all  possible 
values  constitutes  a  group,  if  the  transformation  resulting  from  the 
successive  performance  of  any  two  of  them  is  one  of  the  transforma 
tions  of  the  aggregate. 

As  in  the  case  of  one-parameter  groups  (Chapter  I),  the  groups 
here  considered  arc  supposed  to  have  their  transformations  pair  off 
into  mutually  inrrrse  ones.  That  is,  corresponding  to  -any  set  of 
values  of  alt  a.>,  •••,  <rr  there  must  always  be  another  set  <r},  <7.,,  ••-,  ar, 

*  As  before,  0  and  <//  ;ir»-  supposed  (o  he  <M'ner;illv  analytic  real  functions  of  x,y,  alt 
a-2,  •••,  <ir  :  and,  unless  specially  stated,  it  \\ill  be  understood  that  .r  and  r  are  real 
and  tliat  the  parameters  take  sueh  values  only  as  render  .r,  and  v,  real.  Groups  of 
transt'oi  inations  involving  two  variables  aie  considered  here.  For  the  theory  of  those 
involving  ;/  variables  the  student  is  referred  to  Lie's  works,  especially  his  'rraiisforma- 
tio*jgrttppettt  Vol.  I.,  and  liis  Coiiti/iuit-rUc/ic  (  if  u  />/><•  a  ;  also  to  Campbell's  Introduc 
tory  I  >  <\itis<-  on  /./V.v  Theory. 


212  THEORY   OF   DIFFERENTIAL   EQUATIONS 

(functions  of  the  former  ones)  such  that 


Another  way  of  putting  this  is:  If  the  equations  (153)  are  solved 
for  x  and  7,  the  latter  must  appear  as  the  same  functions,  <£  and  \f/} 
respectively  of  x±  andj^  and  a  set  of  r  functions  of  alt  a2,  •  •-,  ar)  as 
indicated  by  (153). 

Thus,  consider  the  translations 
XVII  xi  =  x  +  ai,  yi  -  y  +  a2. 

If  one  of  them  be  followed  by  a  second  one, 

x-2  =  *i  +  b\y  y-i  =y\  +  bzt 

the  result  is  x%  =  x  -f-  c\,  y«  ----  y  +  c%, 

where  ^i  =  #1  -f  b\,   c-i  = 


Solving  the  equations  XVII  for  x  and  y, 

x  =  x\  —  «  i,  y  —  y\  — 
Hence  a\  =—  a\,    a-i  =—  <*%• 

Again,  consider  the  displacements 
XVIII         x\  —  .rcos  a\  —  jsinai  +  a»*  y\  — 


A  second  transformation  of  this  type 

x.2  =  x\  cos  bi  —  y\  sin  l>\  +  b^  y-i  —  x\  sin  b\  +  yi  cos  b\  +  ^3, 

results  in 

x-2  —  x  cos  ri  —  y  sin  o  +  ro,  jo  =  jr  sin  c\  +  ^  cos  <TI  +  ^3? 

where 

f!  =  rfi  +  ^if    r2  =  a-2  cos  ^i  —  a;5  sin  61  +  b^    CA  =  a»  sin  b\  -f  a?,  cos  ^  -f 

Solving  the  equations  XVIII  for  x  and  y, 
x  =  ^icos(—  tfi)  —  /isin(— 
j/  =  jfi  sin  (—  tfi)  +  Ji  cos(— 


Hence 

i  +  rt;j  sin  </i),   «;$  =  (J-i  sin  <?i  —  a-\  cos  «i. 


APPENDIX 


213 


In  the  case  of  the  general  projective  transformations 

4  a±y  4  a*       —  a*x  4  <*&  v  4 

—  ,   y\  — 


&1*  4-  a%y  +  #9 

there  is  no  difficulty  in  seeing  that  these  constitute  a  group.     For  if  one  of  the 
above  transformations  be  followed  by 


-f 


there  results 
where 


=  «*  +  C^  +  <*  ,       y,  =  <**  +  <*?  + 

^7-^  +  ^8>'  +  ^9  ^7-^  +  C*y  + 


c\  —  a\b\ 


4-  <?7^3, 


=  a-2t>±  4-  rt6*5  4  «8^6, 

—  a36f  4  «o^s  4  <?  9^8, 


^"9  —  «r,^7  +  #6^8  4  ^9^9. 

Moreover,  the  result  of  solving  the  equations  XIX  for  x  and  y  is 

4- 


where  /^i,  /4o, 
determinant 


4  ^e^i  4-  ^9    '       ^3^1  4-  ^c^i  4  - 

are  the  cofactors  of  the    corresponding  elements  of  the 
a\     0o 
a^     «6 


Since  the  successive  performance  of  two  mutually  inverse  trans 
formations  results  in  the  identical  transformation,  the  latter  must 
always  be  a  member  of  the  Lie  group  ;  hence  there  must  always 


214  THEORY   OF   DIFFERENTIAL   EQUATIONS 

i 
exist  a  set  of  values  of  the  parameters,  a",  a.?,  ••-,  <zr°,  such  that 

(I53o) 


It  is  readily  seen  that  for 

XVII,  ai0  =  aj  =  o; 

XVIII,  -tflo  =  «2o  =  fl.,0  =  o  ; 

XIX,  «i°  =  «5<>  =  «9°  =  tfwy  number  (different  from  zero}, 

aj  =  tf3'>  =  «4°  =  rt6°  =  <Z7°  =  tf8°  =  O. 

We  shall  further  presuppose  that  all  of  the  /'parameters  in  (153) 
are  essential,  that  is,  that  the  formuke  of  transformation  cannot  be 
replaced  by  another  set  involving  a  smaller  number  of  parameters 
without  reducing  the  number  of  transformations  represented  by 
them. 

Thus  xi  —  x  -f  ai  +  a3,  yi  —y  +  a-> 

contains  no  transformation  that  is  not  included  in  XVII.     It  involves  only  two 
essential  parameters  ;   a\_  -\-  a?>  is  no  more  general  than  a\. 

In  XIX,  as  is  well  known,  there  are  only  eight  essential  parameters  ;  since 
the  expressions  are  homogeneous  and  of  degree  /ero  in  the  parameters,  it  is  only 
the  ratios  of  the  latter  to  any  one  of  them  that  count. 

A  group  involving  r  essential  parameters  is  known  as  an 
r-paramcter  group. 

It  is  frequently  possible  to  tell  by  inspection  whether  the  parameters 
appearing  are  essential  or  not.  An  analytic  criterion  is  given  by  the 
theorem  of  Note  VII. 

Show  that  the  following  sets  of  transformations  constitute  groups. 
Find  the  respective  values  of  the  parameters  that  give  the  inverse 
and  those  that  give  the  identical  transformations  : 

Kx.  i.   a-j  =  a^x  +  a.2t  )\  =  al  y  +  a* 

Ex.  2.    ,rj  =  a&  +  azy  +  a3t  y\  =  a^c  +  <75  y  +  a6. 

Ex.  3.  ji\  =  x  +  air  +  tfj,  y\  =y  +  «3. 


APPENDIX  215 

Ex.  4.   xl  =  x  +  rt,jr  +  a.>,  yi  =  <r  ..  v. 
Ex.  5.    ^  =  (#!  -|-  i  ),v  +  O,  — 


Infinitesimal  Transformation.  —  The  transformation 


y,  =  f(x,  y,  <  +  8alt  of  +  &»2,  -n  tfr°  +  &*,), 

where  ^^  f72°,  ••-,  «rn  determine  the  identical  transformation  and 
fail,  &*2,  "•>  S^r  are  infinitesimals,  changes  jc  and  y  by  infinitesimal 
amounts,  since  <£  and  ^  are  supposed  to  be  continuous  functions. 
Developing  by  Taylor's  Theorem, 


and  stand  for  what 

da? 


respectively  become  when  al  =  af,  a.2  =  a",  •••,  ar  =  a°,  and  the 
unexpressed  terms  are  of  higher  degree  than  the  first  in  80lf  Sa.2,  •••, 
Bar.  The  changes  in  x  and  ^  are  then 


2l6  THEORY   OF   DIFFERENTIAL   EQUATIONS 

We  shall  suppose  that  at  least  one  member  of  the  pair 


, 

for  each  value  of  /  from  i  to  r,  does  not  vanish  identically,  and  that 
all  of  them  are  finite.  Calling  them  &(#,.}')*  anc^  'ft  (•*».)')  respec 
tively,  the  transformation  may  be  written 


In  exactly  the  same  way  as  is  done  for  one-parameter  groups  in 
Note  I,  it  can  be  shown  that  infinitesimal  transformations  of  the  form 
(155)  always  exist,  even  when  the  parameters  enter  in  such  a  way  that 
for  the  particular  values  a®,  a.2°,  •  •.,  ar°  both  members  of  some  pair 

3«K*,  y,  *°)      dt(x,y,  a0) 


da?  da? 

vanish  identically,  or  if  some  one  of  them  becomes  infinite.  f 

Here  &/b  &a2,  •••,8<zr  are  any  infinitesimal  increments  of  the  first 
order.  Taking  &a  as  a  standard  infinitesimal  of  the  first  order,  we 
may  put 


*  Here  ^  and  77,-  are  written  as  functions  of  x  and  y  only,  since  Oj0,  a.f1,  •••,  ar° 
appear  as  numerical  constants. 

f  The  general  expressions  for  the  coefficients  in  (155)  are,  in  thenotaiion  of  Note  I, 


Here  «lf  «._,,  '••,  «r  are  any  set  of  values  of  the  parameters  for  which  both  t.  and  77* 
are  finite,  and  at  least  one  of  them  is  not  identically  zero.  The  forms  (155)  for  £,,  77,- 
are  what  the  general  forms'  (155')  become  for  the  special  choice  ak  =  (ik  =  at0 
(*=  i,  2,  ...,r). 


APPENDIX  217 

where  tlt  e2,  ••-,  er  are  any  finite  constants.     The  general  type  of  an 
infinitesimal  transformation  may  then  be  written 


^     6x 

v  =  (/IT?!  4-  ^r?, 


For  the  sake  of  brevity  we  shall  write 

(157)  &x  =  £Sti+  •••,  8v  =  i/80  +  —  , 

where  £  =       *<^  an(^  ^  =      ^*"     Intr°ducing  the  symbol 


and  similarly 
we  have 

(158) 

It  can  be  proved  *  that  when  the  r  parameters  of  the  group  are 
essential  Uif,  U^f,--  Urf}  (in  which  £.,  ?;„  are  given  by  (155')  for 
any  properly  selected  set  of  values  of  the  parameters,  in  particular 
they  may  have  the  special  forms  (155)),  are  linearly  independent  ; 
that  is,  that  it  is  impossible  to  find  a  set  of  constants  cly  c.,,  •••,  cr  such 

that  c&f+  cMf+  '-+  crUrf=  o, 

which  is  equivalent  to  saying  that  for  no  set  of  constants  r,,  r,,  •••,  cr 
can  both  the  relations         ,. 


*  Thus,  for  example,  see  Lie,  ContinuierlicheGruppen,  Chapter  6,  or  his  Trans  form  a- 
tioHsgruppen,  Chapter  4.  Also  Campbell,  loc.  cit.,  §  42. 

The  object  of  this  Note  is  to  present  as  compactly  as  possible,  consistent  with  a 
clear  understanding  of  the  chain  of  reasoning,  the  relations  luturrn  /--parameter 
groups  and  their  infinitesimal  transformations.  Consequently  when  long  and  tedious, 
the  proofs  of  certain  facts  are  omitted  here.  These  may,  however,  be  obtained  from 
the  references  given. 


218  THEORY   OF   DIFFERENTIAL   EQUATIONS 

hold  simultaneously.     Moreover,*  it  can  also  be  proved  that  if 
&v  =  E8rt+--.,    8_v  =  H80  +  •.., 

is  any  infinitesimal  transformation  of  the  group,  H  is  a  linear  function 
of  £lf  £2,  •  ",  £r  with  constant  coefficients,  and  H  is  the  same  function 

Of  r,,,  T?.,,  •  •  -,  ,,„,  thllS, 


where  the  set  UJ=  L       +  rli        (/  =  i,  2,  •••,  r) 

dv          c^ 

is  any  linearly  independent  one. 

The  coefficients  of  &z  in  (156)  can  therefore  never  both  vanish 
identically.  Hence  at  least  one  of  the  terms  of  first  order  must 
appear.  Infinitesimals  of  higher  order  than  the  first  may  conse 
quently  be  neglected,  and  the  infinitesimal  transformation  may  be 
written  in  the  form 
(159)  &*  =  (f&  +  t&  +  •••  +  f&)  Stf,  8v  =  (^,17,  +  e,^,  +  ...  -f  erri,)&a. 

The  change  in  any  function  f(^,v}  produced  by  (159)  is  then 

S/EEE  Vf*a, 
where 

(158)  0^4£%/+.'.l%/+  -  +  ^r^r/ 

as  in  §  3,  is  the  symbol  of  the  transformation  (159)  and  will  be  usrd 

to  represent  it. 

The  above  may  now  be  expressed  as  follows  : 

THKORKM  !.•  —  AVvvv  continuous  Lie  ^r<>///>  intNthring  r  essential 
parameters  contains  r  linearly  ini/epeiittenf  infinitesimal  transforma 
tions  (\/\  t\,f,  •••,  frrf,  in  terms  of  which  every  infinitesimal  trans^ 
f(>rmalion  of  tJie  i^nn/p  can  be  e.\  pressed  linearly  until  eons  taut 
coefficients,  tints 

(158)  uf^c,i\f+c.2r,f  +  -  +crurf. 

J/"/'/vv'7'r/-,  fi'frv  transformation  of  flie  type  (i^.for  all"  choices  of 
///e  <'on  slants  eVi  <"._,,  •••  <",.,  l>elt>ugs  /<>  (lie  group. 


APPENDIX 


219 


Remark  i. —  It  follows  that  in  any  set  of  infinitesimal  transforma 
tions  of  the  group,  only  /•  at  most  can  be  linearly  independent. 
Moreover,  starting  with  any  r  linearly  independent  transformations 
UJ,  UJ,  •-,  UJ,  every  set 

VJ  =  *u  1 1/+  <•* UJ  +  -  -  -  +  ^ UJ 

(k=  I,    2,    •••,  /') 

will  be  linearly  independent  provided 


=£0. 


Any  set  of  r  linearly  independent  infinitesimal  transformations, 
V\fi  V-2/>  '"»  Vr/>  may  ^)e  taken  as  the  r  transformations  (referred  to 
in  Theorem  I)  in  terms  of  which  all  the  infinitesimal  transformations 
can  be  expressed  linearly  with  constant  coefficients  ;  for,  since  A  ^fc  o, 
each  of  £/,/,  UJ,  •••,  UJ  is  a  linear  function  of  /',/,  VJ,  ••>,  Vrf 
with  constant  coefficients. 

In  the  case  of  XVII 


A  set  of  linearly  independent  transformations  is 

U\  f—  f      (.'•/' 

~~  dx* 
In  the  case  of  XVIII, 


A  set  of  linearly  independent  transformations  is 


-f  Sa-.\  = 


220        THEORY  OF  DIFFERENTIAL  EQUATIONS 

In  the  case  of  XIX  there  are  only  eight  essential  parameters.     Putting  ag  =  I, 
r  _  „  i  *r  _  (i  +  8<*i)x  +  y  3*2  +  da3 

X\   —  X  -f-   OX  —  -  —  • 

x  da7  +  y  5«  8  -+-  I 
But  -  =  I  —  x  $a-,  —  y  da$  +  •  •  •, 


where  the  dots  stand  for  terms  of  higher  degree  than  the  first. 


...  x\  =  x  4-  8x  =  x  -\-  x?>a\  +  y 

Whence,  5x  =  (e±x  +  e»y  +  €••>,  —  e-jx2  —  e%xy}ba. 

Similarly,  8y  —  (e±x  -\-  e^y^-  e$  —  e-^xy  —  e8}>'2)8a. 


A  set  of  linearly  independent  transformations  is 


Ex.  7.     Find  the    infinitesimal  transformations  of  the  groups  in 
Ex.   i,   2,  3,  4,  5,  6  above. 

Group  Generated  by  Infinitesimal  Transformations.  —  Starting  with 
the  infinitesimal  transformation 

(158)  67  =  ^7+^7+  ...  +eruj 


in  which  the  constants  elt  e^  ••«,  er  are  fixed,  the  finite  transforma 
tions  of  the  group  generated  by  it  may  be  obtained  either  by  finding 
those  solutions  of 
(i  60) 


APPENDIX  221 

for  which  xl  =  x  and  yl  =  y  when  /  =  o  (§  4),  or  in  the  form  (§  5), 

£ 
(161) 


In  both  cases  /  is  the  parameter,  and  /  =  o  gives  the  identical 
transformation. 

If  elt  e2,  •••,  er  are  arbitrary  constants  and  c^/,  £/,/,  •••,  U/  are 
linearly  independent,  the  infinitesimal  transformation  contains  r— i 
parameters  (viz.  the  ratios  of  any  r—  i  of  the  <?'s  to  the  remaining 
one),  and  the  general  expression  (161)  for  the  finite  transformations 
generated  by  it  contains  r  parameters.  That  these  parameters  are 
essential  follows  from  the  linear  independence  of  £/,/,  c7,/,  •••,  £//. 
A  proof  of  this  fact  may  be  found  in  Lie's  Continuierliche  Gruppcn, 
pp.  186-190.  Hence  there  are  ocr  transformations  in  the  set  (161). 

If  £/i/",  U%f,  -",  6//are  r  linearly  independent  infinitesimal  trans 
formations  of  an  r-parameter  group,  every  transformation  of  the  set 
(158)  belongs  to  the  group  (Theorem  I).  All  the  transformations 
of  the  one-parameter  group  generated  by  any  transformation  (158) 
belong  to  the  r-parameter  group  (Lie,  Continuierliche  Gruppcn, 
p.  183).  The  ocr' transformations  (161)  therefore  belong  to  the 
group.  Moreover,  every  transformation  of  the  r-parameter  group 
(at  least  all  such  for  which  the  values  of  the  parameters  are  suffici 
ently  small  so  that  when  developed  by  Taylor's  Theorem  in  powers 
of  the  parameters,  as  (161)  are,  the  series  are  convergent)  is  in 
cluded  in  (161)  (Lie,  Transformati0nsgrupp€itt  Vol.  I,  Ch.  4,  §  18). 
Hence 

THEOREM  II.  If "  U\f,  U»f,  •••,  UJare  r  linearly  independent  trans 
formations  of  an  r-para  meter  group,  the  latter*  is  precisely  the  OggTV- 

*  At  least  all  its  transformations  corresponding  to  values  of  the  parameters  which 
differ  by  limited  amounts  perhaps  (see  above)  from  those  which  give  the  identical 
transformation. 


222  THEORY  OF  DIFFERENTIAL  EQUATIONS 

gaff  <>/  all  the  one-parameter  groups  generated  by  the  cc7—1  infinitesimal 
transformations 


Remark  2.  Since  /  and  the  <^'s  appear  in  (161)  in  the  combina 
tions  /<?!,  /if2,  •••,  ter,  there  will  be  no  loss  in  suppressing  the  /  and 
writing  the  finite  transformations  of  the  group  in  the  form 


(161') 


I 

2~! 


where  the  <r's  are  now  r  distinct  parameters.  The  identical  trans 
formation  is  given  by  el  =  e.2=  •»=er  =  o,  and  the  inverse  trans 
formation  by  ei  ;  =  —  ci  (/=i,  2,  •••,  r). 

In  the  case  of  XVII  the  general  type  of  infinitesimal  transformation  is 


The  finite  transformations  (161')  are  seen  at  once  to  be 

Xi  =  X  +  «?!,  jj'i  =^  -f  t». 

In  the  case  of  XVIII 


.'.  #i  =  x  —  o^  4-  ^  -  -1-  e{-x  +  -1-  epy  +  --*!4jr  —  ••• 

=  x  cos  <TI  —  y  sin  <?  i  -f  *••_>. 
Similarly  ^'!  =  JT  sin  e\  -f-  ;'  cos  <TI  +  ^3. 

Remark  j.  The  expressions  for  jr,  and  r,  in  (161')  may  at  times 
become  extremely  complicated,  as  for  example  in  the  case  of  the 
group  XIX.  Also  the  actual  problem  of  integrating  equations  (160) 
with  the  r's  arbitrary  constants  is  usually  a  difficult  one.  To  over 
come  this  practical  difficulty  T,ie  suggested  the  following  method, 
which  was  also  given  independently  by  Maurer  (  Matli.  ./////.,  Vol.  39)  : 


APPENDIX  223 

Having  found 

(162)  *}.  =  4>i(x,  y,  <O»  yi  =  <l>i(x>y>ai)> 

(/  =  i,  2,..-,  r), 

the  finite  transformations  of  the  one-parameter  groups  generated  by 
each  of  the  r  linearly  independent  infinitesimal  transformations 
U\ft  U.>f,  -•-,  £7r/of  an  /--parameter  group,  the  result  of  performing 
successively  one  transformation  (with  arbitrarily  selected  parameter) 
out  of  each  of  the  ;•  groups  (162)  is  a  transformation  belonging  to 
the  r-parameter  group  and  involving  the  r  parameters  alt  a.,,  •••,  ar. 
That  these  are  essential  follows  also  from  the  linear  independence  of 
t  Urf.  (See  Lie,  Continuicrlichc  Grnppcn,  p.  194.) 

In  the  case  of  XVII 

=& 
flbr 

U*f=  (f  :    u-g  =  xlt 

By 

The  successive  performance  of  these  gives 

*z  =  x  4-  rti,  y»  =  y 
In  XVIII 

rr  f  <V   ,       3/ 

t/i/=  —y  -^  4-  •«•—  :  -*i  = 

=-  = 


The  successive  performance  of  these  gives 

jc-.\  —  A  cos  <zi  —  _j'  sin  </i  +  </;.,  _j';!  =  x  sin  ^/i  +  7  cos  ^i  4-  r/:;. 
In  XIX 

U\f=(.      :  xi=x  +  rt|,      Vi  =  j-,  , 


224  THEORY   OF   DIFFERENTIAL    EQUATIONS 


~dy  ' 

df 
Usf=y  £  :  x*  =  xb,  y&  =  ^5, 


dx          dy  i  -  «8j'r 

The  successive  performance  of  these  gives 
x  _ 

_ 
where  <4= 


Find  the  finite  transformations  generated  by  the  following 
Ex.    8.    U= 


oy 
Ex.    9.        «  *  +  **  +  *i^  +     +'t 


Ex.10.    &&(<&  +  *        +  €. 

ox         oy 

Ex.  11.    Uf=  (etf  +  ^2)  I"-  +  'V'f  . 
3jc  61)' 

Ex.  12.    £/EE  (^  4-  e«x  H-  <,  v)  ;X  +  (', 

(U 

Ex.  13.    ^-  (^  +  ^2^  4-  W)      4 


APPENDIX  225 

Lie's  Principal  Theorem.  —  It  was  shown  above  (Theorem  II) 
that  if  £/!/,  £/,/,  ...,  UTf  are  r  linearly  independent  infinitesimal 
transformations  of  an  r-parameter  group,  the  aggregate  of  the  cor 
transformations  of  the  ccr~l  one-parameter  groups,  each  generated 
by  an  infinitesimal  transformation  of  the  set 

(158)  Uf  =  f  \Uif-\-  e^U-J -\-  •"  +  eTUrf, 

forms  an  r-parameter  group.  On  the  other  hand,  starting  with  any 
r  linearly  independent  infinitesimal  transformations  L\f,  U.J.  •••, 
Urf  (without  knowing  whether  they  form  a  complete  set  for  some 
group),  there  is  no  reason  to  suppose  that  the  oor  transformations 
generated  by  the  various  transformations  (158)  form  a  group. 

Thus,  starting  with  £/I/=^T,    EV=*f£' 

CM  vy 

the  transformations  generated  by 

_-.- of  of 

LJf  =:  e\  — — \-  f-~>x  -7— 
dx  dy 

are  xi  =  x  -f  a\,  y\  =  a»x  +  y  +  et-^- 

While  these  transformations  involve  two  essential  parameters,  it  is  very  easily 
seen  that  they  do  not  form  a  group. 

A  definite  answer  as  to  when  the  oor  transformations  generated  by 
the  various  transformations  of  the  set  (158)  form  a  group  is  given  by 
LIE'S  PRINCIPAL  THEORKM  :  *  The  necessary  and  sufficient  conditions 
that  the  oor  transformations  generated  by  the  ocr~1  infinitesimal  trans 
formations  TJ  f  +-  TJ  /  -U  -4-  Tr  f 

*  Lie  calls  this  theorem  "  Der  Hauptsati  der  Gruppentheoric,"  and  gives  a  proof 
of  it  for  groups  involving  two  variables  in  his  (\wtinuifrlichf  Grtippen,  Ch.  12.  In 
his  treatment  of  the  general  theory  of  continuous  groups,  this  theorem  is  the  second  of 
his  "three  fundamental  theorems."  See  his  (\>ntinuierliche  C,ruf>f>en,(Z\\.  15,  or  his 
Transformation wupp.'n.  Vol.  I,  Ch.  9 ;  also  Campbell,  /,»r.  cif.,  Ch.  IV. 

A  detailed  proof  of  this  theorem  would  be  beyond  ihe  scope  of  this  Note.  A  state 
ment  of  it  with  illustrative  examples  will  suffice. 

Lie  first  deduced  this  theorem  in  1874. 

Q 


226  THEORY   OF   DIFFERENTIAL   EQUATIONS 

where  U^f,  U-,f,  •",  UJ  are  linearly  independent  and  <?lf  e.2,  •  ••,  er  are 
any  constants,  constitute,  a  Lie  r-parametcr  group  arc  t/iat 


(/,£=i,  2,  ..-,  r), 
where  the  c's  are  constants. 

Remark  4.  —  This  theorem  is  equivalent  to  the  following  two  : 

i°.    TJie  infinitesimal  transformations  of  an  r-parameter  group  form 

tin  r-parameter  group  of  infinitesimal  transformations.      (§  43.) 

2°.    27u'  transfoi-mations  if  tlic  groups  generated  b\  tJie  transforma 

tions  of  an  r-para  meter  group  of  injin  itcsimal  transformations  form  an 

r-parameter  group. 

In  the  case  of  XVIII, 


Here   (l\U*)f=Uzf,    (U,U^f~-U^ft    (U,U*)f=o. 
In  XIX 


(  ^,  £/7y=  2 

and  so  on. 

Ex.  14.     Show  that  the  infinitesimal  transformations  in  Ex.  8  to 
13  satisfy  the  conditions  (163). 

NOTE   VII 

CONDITION  FOR  ESSENTIAL   PARAMETERS 

The  r  parameters  in 
(J53)       -vl  =  <#>(.v,.v,  a},  a.,,  ••-,  ar),    )'1  =  t//(.v,  r,  alt  a,,  --,  ar) 

are  //^/essential  if  (153)  <~an  be  n-nlared  by 

(164)      j;1  =  *(jr,^,  «!,  «,,  ..-,  «r_TO),  .Vi  =  ^Cv,.v,  «i,  «,.,  •••,  «r_m) 


APPENDIX  22/ 

In  this  case  the  identities 

(165)  <£  =  <!>,    ^  =  * 

for  all  values  of  x  and  y,  determine  alt  a2,  •••,  ar_m  as  functions  of 
a\>  a-2,  -~,  (iT  ;  f°r  by  saying  that  (164)  replaces  (153)  is  meant  that 
as  soon  as  the  #'s  are  given  a  set  of  the  a's  is  determined  (not  neces 
sarily  uniquely)  which  will  give  rise  to  the  same  transformation. 

A  homogeneous  linear  partial  differential  equation  of  the  first  order 
in  r  variables 

(166)  4/~Xl(alf  a,,  •••,  ar)---f--\  -----  hXrOi>  «z,  —j  <*r)~-  =  o 

oa1  dar 

is  determined  uniquely  by  r—  i  independent  solutions.*  An  equa 
tion  of  this  type  can  therefore  be  constructed  which  shall  have  for 

solutions 

«!,  «,,  •••,  «r_m,  /?r_m+1,  •••,  (3r  „ 

where  /3f_m+l,  •••,  /?r_],  any  convenient  functions  of  the  #'s  inde 
pendent  of  the  «'s,  are  added  to  the  latter  to  make  up  the  number 
r—  i  in  case  ;//  >  i.  This  equation  will  have  for  solution  also  any 
functions  of  the  «'s,  in  particular  $  and  >P,  x  and  y  appearing  as 
parameters  ;  or  owing  to  the  identities  (165),  by  which  the  «'s  are 
defined,  <f>  and  ^  will  also  be  solutions. 

Conversely,  if  <£  and  \j/  satisfy  an  equation  of  the  type  (166),  they 
are  functions  of  some  or  all  of  its  r  —  i  solutions. 


i.e.  the  #'s  enter  <#>  and  \f/  in  such  a  \vnv  that  for  all  values  of  A*  and  y 
<£(.v,  v,  <7.,  t7...  •••,  ^7,)  =.1  ^(.v,  r,  y,,  y,.  •••,  yr_0, 
^(.v,  i',  ^,.  <A_,,  •••,  ^7,)  =  M'(  v.  r,  y,,  y,,,  •••,  y,^). 

*  A  proof  of  this  for  the  rasr  of  r      3  is  i^ivcn  in  tlu'  in  -t  footnote  of  ij  34.      The 
proof  for  r  any  number  is  exactly  the  same. 


228  THEORY   OF   DIFFERENTIAL   EQUATIONS 

Hence  the 

THEOREM.  —  The  necessarv  and  sufficient  condition  that  the  r para 
meters  in  (153)  be  essential  is  the  impossibility  of  Jin  ding  r  functions 
of  them  xu  Xt>  '"•>  Xr  Sltc'h  that  the  resulting  linear  equation  (166) 
shall  have  <£  and  \\i  for  solutions. 

Remark.  —  There  is  nothing  in  the  above  to  show  whether  the 
r—  i  parameters  y1?  y2,  •••,  yr_:  are  essential  or  not.  The  same  test 
must  be  applied  to  them  also,  unless,  as  is  frequently  the  case,  the 
exact  state  of  affairs  is  obvious  on  inspection. 

To  illustrate,  consider  the  transformation 

xi  =  xa]°sb  -f  #<*«  -1-  c  =  <}>(x,  y,  a,  b,  c), 
yi  =jj/«log6  =  \f/(x,  y,  a,  b,  c}. 

If  a,  l>,  c  are  not  essential  it  must  be  possible  to  find  three  functions  of  them, 
Xi(fl,  b,  f),  X2(X  b,  c),  X$(a,  b,  <r),  such  tliat  the  equation 

(i  66)  Af=XM  +  x>M+x,V=o 

da          db          Qc 

is  satisfied  identically  (for  all  values  of  x  and  jy)  by  0  and  ^;   that  is 


,=1^(^108*  + 

a 


\    a  b 

for  all  values  of  x  and  y.     These  two  identities  are  equivalent  to 

=  o, 


=  o. 


APPENDIX  229 

By  inspection,  a  set  of  forms  for  xi»  X2»  %3  are  found  to  be 


Hence  the  three  parameters  are  not  essential. 

To  express  the  formulae  of  transformation  in  terms  of  a  smaller  number,  one 
proceeds  to  solve  the  equation 

(166')  a\oga&-  b\ogt>i¥=o.  * 

da  fib 

Passing  to  the  corresponding  system  of  ordinary  differential  equations 

da  db        _  dc^ 

a  log  a       —  b  log  b       o 

it  is  obvious  that  log  a  log  b  and  c 

are  a  set  of  solutions  of  (166').     Putting 

log  a  log  b  —  «,  whence 

the  formulae  of  transformation  take  the  form 


or,  more  simply  still,          ^i  =  a^  -f 


TABLE    I 

IN  this  table  is  given  a  list  of  the  more  readily  recognizable  forms* 
of  differential  equations  of  the  first  order  which  are  known  to  be  in 
variant  under  certain  groups.  The  same  type  of  equation  is  some 
times  given  in  various  forms,  and  special  cases  are  also  noted  when 
this  seems  desirable. 

In  the  second  column  appear  the  groups  under  which  the  equations 
are  invariant.  The  numbers  are  those  employed  in  §  19.  For  the 

sake  of  simplicity  /  and  q  are  used  instead  of     -  and  J.  respectively. 

dx  oy 

The  corresponding  integrating  factors  of  §  12  are  given  in  the 
third  column. 

In  the  fourth  column  appear  the  canonical  variables. f 
When  variables  which  are  separable  in  the  transformed  equation 
(§  20)  can  be  obtained  easily,  they  are  given  in  the  fifth  column; 
the  form  of  the  group  resulting  from  the  introduction  of  these  vari 
ables  is  given  in  the  last  column. 

*  Other  forms  will  be  found  in  $  19. 

t Then-  is  :i  certain  dearer  of  freedom  in  the  choice  of  cnnonie;il  variables,  since 
they  are  particular  solutions  of  the  differential  equations  (i^'l.  ^  10,  or  of  the  eorrcspond- 

d/' 
Ing  ones  in  case  the  group  is  to  be  reduced  to  the  form  ,   •    Moreover,  the  right-hand 

member,  i,  in  one  of  these  equations  may  be  replaced  by  any  convenient  constant  (see 
Remark  i,  $  2)  ;  use  of  this  fact  is  made  when  it  will  simplify  the  form  of  the  resulting 
variable. 


231 


232 


THEORY  OF   DIFFERENTIAL   EQUATIONS 


HI 


*   £ 


^    a 


^a 

u  - 

52 


-f 


I 


- 


o"      w 

ii     § 


o 

II 

/-~s 

X 

o 


"V 


V 


TABLE   I 


233 


ANO 

ARI 


II     II 


- 


t 


X 


"\    8    v. 

•ft1* 

g    "     " 

*&  * 

</i    •"     ti 
3    £    c 


s 


J-    ?      >s 

+  !r  + 


> 


^    B.     *S 
+     C      + 


234 


THEORY  OF  DIFFERENTIAL  EQUATIONS 


«  -f 


^ 


EQ 


•2  ^ 

rt   . 
^J 

^ 


. 


_ 

«j  a^  "3  1 

+  1 


TABLE    I 


235 


-H 


-H       -i 


H         * 

i  a 


'H      0      C 

: 


. 

i      ,2    H   ^ 


K- 


H- 

. — •• 
1 1  -e- 


r 


o 
II 
^ 

+ 
? 


»*\ 

-H 


-H 


;  s 


i          i 


^    ^ 

i      i 


TABLE    II 


General  types  of  differential  equations  of  higher  order  than  the 
first,  invariant  under  given  groups,  are  usually  complicated  and  not 
easy  to  recognize.  In  this  table  are  given  a  few  which  a  little  ex 
perience  will  enable  one  to  recognize.*  Such  characterizations  as 
are  simple  are  added. 


DIFFERENTIAL  EQUATION 

GROUP 

CHARACTERIZATION 

/(*,/,/',  -,yW)  =  o 

i;  q 

y  is  absent 

/(y,y',y",-,y^=o 

i';/ 

.r  is  absent 

f(ax  +  by,y',y",  ...  ,  ;/(O)  =O 

XII;    bp-  aq 

x  and  jy  enter  in    the    combi 
nation  ax  -f  by  only 

iiii-fh 

HI;  yq 

Homogeneous  my,y',  •••,  y(r) 

f(y    y>       y^\  o 

VI  ;  xp  -f-  nyq 

Homogeneous    \\hen   weight! 
of  x,  y,  y1,   •--,  \>(r)  an-  ',  n, 

'U»'  *«-«'        '  xn-r) 

n  —  I,  •••  ,  n  —  r  respectively 

/K  /.*/',-, 

\x                                            . 
xr-iy(r)  J=0 

IV;  xp  +yq 

Special  case  of  above,  for  n  —  i 

f(y,  xy',  x2y",  ...  ,  x*y(r))=o 

III';   xp 

Another  special  case,  for  n  =  o 

f(x,<t>y'-<t>'y,<t>y"-<t>"y,   ..., 
<f>y(r)  —  $(r}y}  =  Q 

VII;    0O)? 

A  linear  function  of  the  vari 
ous  elements,  except  x,  gives 
rise    to    a     linear   differential 
equation 

ftxtxy'-*y,*?y"-*(k-i)y, 

••-,  xry(r)—k(k-i)... 

(k-r+i)y~\=o 

VII;   ^ 

A  special  case,  for  0(.*)  =  x& 

*  Other  forms  will  be  found  in  §  28. 


TABLE   II 


237 


DIFFERENTIAL  EQUATION 

GROUP 

CHARACTERIZATION 

f(*,*y'-y.  /',/",•••, 

gy(r))=o 

VII;    xq 

A  more  special  case,  fur 

0O)  =  * 

/      y              y"y      s—l\ 

/('v-"'>  +  7-)=0 

VII';  ysp 

• 

/H-^H 

VII';  j/ 

/(*,.>y,;'/'+/2)=o 

VIII;  ^ 

/fe  */-;•»  *y')  =  o 

X  ;    ^-2/  +  xyq 

/Oj,  *•/  +,v,  xy"  +  2/)=  o 

X;/-^y 

,fr,,,2  izs!     ym  \ 

II;    -^/+^ 

Each  of  the  elements  appear 
ing  in  the  differential  equa 
tion  has  a  geometrical  signifi 
cance,  which  assures  invari- 
ance  under  the  group  of  rota 
tions 

'V    l}'  x+yy»  (i+/2)3J 
=  o,  or 

f(#++  *+yy'      y"'2  \ 

f\    l}'  Vi+y^'Cu-y^J 

fa+Aj^r-j!*^) 

\           Vi+/^  '(1+y"?) 

=  0 

ANSWERS 

Section  1 

1.  a  =  -  ;   ao  =  I  ;    the  equilateral  hyperbolas  x\y\  —  xy  =  const. 

a 

2.  a  =     ;   dQ  =  i  ',  the  parabolas  •?—  =•  ^—  —  const. 

3.  a  =  _  ;    fl0  =  i  j    the  semicubical  parabolas  *-*-  = «—  =  const. 

(i  x\^      x^ 

4.  tf  =  —  «  ;   rto  =  o  ;   the  ellipses  x\*  -\-  2y\*  =  x'*  +  2^v2  =:  const. 

5.  #  =  —  «  ;   ao  =  o  ;    the  equilateral  hyperbolas  xfi  — y\*  =  x^  — y*  =  const. 

)'i        v 

6.  a  =  —  a  •   ^  =  o  ;   the  straight  lines  —      ~  =  const. 

7-    a  =  — 'y   ao  =  I ;   the  straight  lines  ^i  =  jy  =  const. 

8.    fl  =  —  a  ;   ao  =  o  ;   the  spirals  log  \/.rr  +  J'i"  —  tan-1  ^1 


=  log  vV2  +  y2  —  tan-1  £  =  ^«j/, 


Section  2 

•*'  2>/  7-   f  = 

5.   |=;',   >n  =  x. 

Section  4 


4.    x?  -  .v-  +  2  /,  ^!2  =;'2  -  /.     .-.*!=+  Va-^  +  2  /,   j/i  =  +  Vy2  —  /. 

5-    -VL  +jJ'i  =  ^ 


or 


.n  =  JT  cosh  t  +  y  sinh  /,  ;'t  =  x  sinh  /  +  v  cosh  /. 


6.        »,   -=_,.    ...  ^=_^L_,  ^l  =  ^L_. 

.^i       x      •>  i       Jf  I  —  .v/     '          I  -  .r/ 


7-   .''i  =J,    .r,  +^i  =  **(jr  +  V).      .'.   xi-f*x-\r(et—  l).r,    v,  =jj/. 

8.    .vrl.rr       <'-'(.\-   '    i-'-',    tun   '  -1'1  =  tan"1  ^  -f-  /.    .'.  :ri  =  «f(.vct>s/—  j-sin  /), 


240  ANSWERS 


Section  5 

2.    xi  =  e*x,  y\  —  (fy.         3.    XY  -  e-'x,  y\  = 


*  2  !      x5  3  !        x{    4  ! 

l«  il'i>.!ii 

4/2!      8^3!       16/4! 


-f  _>'  sinh  /. 


/  /3  /5  \  /  /-  /l  \ 

=  .*•/  +  --  1  ---  1-  ...  \+  y(  i  -|  ---  1  ---  \-  •  ••}  =  jc  sinh/  +  v  cosh/. 
V        3  !       5  !  /A         2  !      4  !  / 


6.  *!=* 

I  —  tX  \—tX 

7.  jT!  =  tx  +  (^  -  i>,  yi  =  y. 

8.  While  the  coefficients  in  the  developments  can  be  obtained  readily,  it  is  not 
easy  to  recognize  the  functions  represented  by  the  infinite  series. 


6.    **•        7.  y. 

X 


Section  7 

1.  xy  =  f,  p.  c.,*    x  =  y  =  o,  i.  p.  6.  y  =.  ex,  p.  c.,   x=  o,  l.i.p. 

2.  j'2  =  ex,  p.  c.,    ^r  =  y  =  o,  i.  p.  7.    r=  c,  p.  c.,  x  +  y  =  o,  1.  i.  p. 

3.  y2  =  fx-\  p.  c.,   x  =  y  =  o,  i.  p.  8-    jog  v^a^jTyj  _  tan-i  ?  -  ^  p>  Ct> 

4.  y2  -f  2y2  =  c,  p.  c.  ^  =  y  =  o,  i.  p. 

5.  x2  -y2  =  r,  p.  c.,   x  =  y  =  o,  i.  p. 

Section  10  f 

3.  *=•£  r=kg* 

»" 

4.     *==*•+  2;"-,     i/=/. 

*  The  abbreviations  here  used  are:  p.  c.  for  path-curve,  i.p.  for  invariant  point, 
1.  i.  p  for  locus  of  invariant  points. 

t  The  answers  ^ivni  lor  the  exercises  of  this  section  are  not  unique,  since  they  are 
particular  solutions  of  tin-  differential  equations  (16').  Besides,  the  right-hand  member 
of  the  second  of  these  equations  may  be  replaced  by  any  convenient  constant  (see  Re 
mark  i,  $  2)  ;  use  of  tins  fact  has  been  made  in  the  case  of  Ex.  3,  4,  6. 


ANSWERS  241 


Section  11 

I.    Uf=*=£\   x  =  a,  y  =  t>,  p.c.;*  2  =  0,  l.i. p.;    x  =  x,    y  =  y,    z  =  logz, 

'       >  y  —  ax,    z  =  b,    p.  c.;    x  —  y  =  o,    l.i.  p.;    x  =  ten~l-¥- , 
dy  x 

+  r  ,    z  =  2,  c.  v.     .'.  x  —  e^  cos  x,  y  =  $  sin  x,  z  =  z. 

3.    Uf=  x &.  -\-yty-  -f  Z^L  ;   z  —  ax  =  o,  z—  by=o,  p.  c.;   x  =  y  =  z  =  o,  i.  p.; 
d-*        dy       dz 

x  =  tan"1 —  —  =  tan-1 —         — ,  y  =  tan   1-r  ,  z  =  logV^r2  +y2  +  2^,  c.  v. 

.-.  x  =  ez  cos  AT  cos  y,  y=ez  cos  JT  sin  y,  2  =  ez  sin  jr. 

The  introduction  of  polar  coordinates  reduces  the  group  to  the  form  of  the 
group  appearing  in  Ex.  I. 


4.    Uf=  x-  +  y=-  +  xy--\  y—  ax  =  o,  xy—  2  z  =  b,  p.  c.;   x  =  y  —  o,  1.  i.  p. 
d*         dy  dz     ' 

x  =  tan-1  2  ,  y  =  xy  —  2  z,  z  =  log  vV2  +y2,  c.  v. 


dy       dz  x 

—  bz^  =  o,  p.  c.  ;   x  =  y  =  z  =  o,  i.  p.  ;   x  —  u\y  y  =  u»,  z  —  log  z,  c.  v. 

Section   12 


2.  x2  -f  y-  —  cy  =  o.  3.    tan"1-^  =  Vx*  +y2  +  c,  spirals  [p  =  6  +  <:]. 

-f  ^ 

4.  tan"1  -^  =  /^  logWJ  -f  y2-  +  c,  logarithmic  spirals  [p  =  ce~*~\. 

5.  x'2  +  j-  —  r^r  =  o. 

Section  21 

3.  xy  =  c*x  +  c,  g.  s.,t  4  jr2_y  +  I  =  o,  s.  s.,  y  =  o,  p.  s.  for  c  —  O  ;    §  25,  5.  J 

*  The  abbreviations  used  in  the  answers  of  \  7  are  also  employed  here,  with  the 
additional  one  c.  v.  for  canonical  variables. 

f  The  abbreviations  used  IKTC  arc  g.  s.  for  general  solution,  s.  s.  for  singular  solu 
tion,  p.  s.  for  particular  solution. 

J  While  the  methods  of  $$  12  and  20,  especially  the  latter,  may  frequently  be  employed 
in  finding  the  general  solution,  serious  practical  difficulties  may  arise.  The  references 
he*re  given  are  to  the  places  in  El.  Dif.  Eq.,  where  these  differential  equations  appear 


242  ANSWERS 

4.  ii'ty-  +  2  c x  +  c -  =  o,  g.  s.,  x1  —  a-y-  =  o,  s.  s.,  y  •=.  o,  p.  s.  for  c  =  o,  §  27,  8. 

5.  y  —  c(x  -  c-y2,  g.  s.,  y(2~i y  -  4-rJ)=  o,  s.  s.,  also  j  =  o,  p.  s.    for  c  =  o; 
§  26,  4 

6.  y2  =  2  ex  +  r»,  g.  s.,  (32  JT*  +  27^)  =  o,  s.  s.,  ^  =  o,  p.  s.  for  c  -  O;   §  27,  7. 

7.  JT  4-  ^/  +  ^  =  o,  g.  s.,  jry-  -  4  =  o,  s.  s.;    §  28,  3. 


Section  24 

1.  The  equilateral  hyperbolas  x*  —  y-  =  c. 

2.  y"  =  cxb.     .-.  y  =  ex,  when  b  =  a  ;   xy  =  c,  when  b  =.—  a. 

3.  The  circles  .*•-  +^2  +  I  =  <r^r. 

Section  26 


.  , 

'    d*        dy'  dy" 

3.    vw/=x&-y&-2y'& 


4.    £/(»>/=  ^       4-  by-+  (l>  _  a)yi+(t  -  2  a)y"--- 
3-f         dy  dy'  dy" 


as  exercises.     Practicable  methods  may  be  found  tin-re.     But  when  the  methods  of 
the  text  can  be  carried  on),  tlu-y  should  be  employed,  to  obtnin  practice  in  them. 

However,  the  method  of  ^  21  for  finding  the  singular  solution  leaves  nothing  to  be 
desired.     (Compare  El.  Dif.  /•:,/.  Chapter  V.) 


ANSWERS  243 

Section  23 

/o       N 

3.   y=axe*.  4.   y  =  x  log  —  +  c»  }•  5.  y  =  x(c^  log*  +  <%,). 


Section  29 


Section  32 


3.   Ar2+jj/2.       4.  5.  —  -        6.   jr2  +/2  +  82-  2yz-2zx-2xy. 

z  cx—az 


7-      ^  _ 

Section  34 

The  group  I  leaves  a  and  </  unaltered. 

The  group  2  leaves  a  unaltered. 

The  group  3  leaves  c  unaltered. 

The  group  4  leaves  a,  b  and  c  unaltered. 

Section  35 

3.  u=y  —  x,   v=(x  +y)(x  +y  +  42),  or  xy  +  yz  +  zx. 

4.  u  =  y  +  *  +y~+-xz,    v  =y  —  x  -  yz  +  xz,  or  it  =y  +  xz,    v  =  . 

y-X-yZ  +  XZ 

Section  38 

3.    Method  A,  3°  applies.  4.    Method  B,  4°,  O)  applies. 

5.  Method  B,  i°  applies.     u  =  x  —  y,   v=y  —  z. 

+  xz 

+  yz 


6.    Method  B,  3°  applies.     u  =  ?  +  *Z ,   v=(x-  -/-)(!  —  c3). 


Section  39 

3-  y  =  log  sec  (x  -f  a)  +  b.  4;  y2  =  ax*  +  bx. 

Section  40 

5.   y  —  log  sec  (j:  +  tf)-f  //.  6.    X   ~c\<r.  8.    <*  =  ax*  + 

J' 


244  ANSWERS 

Section  44 
i.    «.         2.    5.         3.    /3.         4.    /3.          5.    7.         6.    7. 

Section  45* 

i.   JT=tan~1-^,    y  =  tan"1  <^  +  log  \/ 


X 

2.    Since  (UiU«)f=  (/if-  U»f,  consider  Ft/=  67i/"-  Uzf=y&-  and  F2 
5tf,/3E*|£.    For  these*  =j,  y=*. 


4.   JT  =     ^  -f  y8,  y  =  tan 


=       -1 


+y         x+y  x 

6.   Since    (UiU^f=l\f,    consider    Vvf=  U«f,    F2/=  -  U\f.      For   these 

Section  49 

i.   x\  =  x  —  ap,  ji  =  y  —  \  ap2,  p\.  —  p. 

Vfl*/2  +  IP     '  \/a^~+lfl' 

3-  -*1!  =  —7*  .   ,   ->  ^i  =  ^T-T~T»  /i  -  7.    -m — rrs* 


NOTE   VI 


2.      1=,,=  —  -,     3= 


A  A  A  A  A  A 

A  =  aids  -  a«cn  ;   a{°  =  a^  =  i,   «a0  =  <z3°  =  a4°  =  <z6°  =  o. 

3.    ai  =  —  a  i,    «o  =  —  0o  +  ^i«:5,    «;}  =  —  #u  J    «i°  =  02°  =  tf:*0  =  °- 

*  Since  multiplying  its  symbol  by  a  constant  does  not  affect  the  infinitesimal  trans 
formation  of  a  group  (Remark  i,  $  2),  the  answers  in  this  section  are  not  unique.  Use 
is  made  of  this  fact  in  Ex.  i,  3,  4,  6.  %* 


ANSWERS  245 

-     -    I      —    _  a>i      -   _       a% .        0 


o-*  oy 

U,f=  („,  +  e,y  +  „)  £.  +  (,„  +  «,,  +  «,) ^. 


i/6/=  Ol 

The  groups  generated  l>y  the  infinitesimal   transformations  of  Ex.  8  to  13  are 
precisely  the  respective  groups  of  Ex.  I  to  6. 


INDEX 


The  numbers  refer  to  pages. 

The    following    abbreviations    are    used  :    dif.    eq.  EE  differential  equation  ;      gr.  —  group 
infl.  EE;  infinitesimal  ;    i.  u.  EE  invariant  under  ;    ord.  EE  order  ;    tr.  EE  transformation. 


Affine  tr.,  3,  54 
Alternant,  44 

of  symbols  of  extended  trs.,  209 
Asymptotic  lines,  80 

Bernoulli  equation,  58 

Canonical  form,  26,  34,  64,  155 
Canonical  variables,  26,  34,  64,  156 
Change  of  variables,  23,  33,  188 
Characteristic  function  of  infl.  contact  tr., 

186 

Classification  of  two-parameter  grs.,  152 
Commutator,  45 
Complete  system,  104,  106,  no 

equivalent,  106 

Jacobian,  107,  no 
Contact  tr.,  178,  181;  infl.,  185 
Curvature,  lines  of,  81 
Curve  of  union  of  elements,  175,  189 

Differential  equation  of  I.  ord.,  189 
i.  u.  gr.,  40,  44,  45,  46,  48,  50,  52,  194,  231 

Dif.  eq.  of  2.  ord.  i.  u.  gr.,  86,  90,  134,  137, 

148,  165,  236 
not  i.  u.  any  gr.,  206 

Dif.  eq.  of  n.  order  i.  u.  gr.,  99,  101,  236 

Differential  invariant,  51,  88,  194 

Dilatations,  185,  186 

Displacements,  212 

Distinct  grs.,  122,  123,  125 

Distinct  infl.  trs.,  7 

Elements,  lineal,  175 

union  of,  175,  194 
Equivalent  complete'  systems,  106 
Kssential  parameters,  214 

condition  for,  226 


Extended  gr.,  42,  84 
Extended  point  tr.,  41,  83,  180 

First  differential  invariant,  51,  194 
First  integral,  191 

General  expression  for  gr.  leaving  dif.  eq. 

of  i.  ord.  unaltered,  49 
Group,  i,  28,  2ii 

distinct,  122,  123,  125 

extended,  42,  84 

generated  by  infl.  tr,,  10,  12,  14,  30,  220 

involving  one  parameter,  i,  28 

involving  /  parameters,  211 

of  contact  trs.,  185  . 

of  infl.  trs.,  146 

property,  2 

trivial,  39,  119,  196 

Homogeneous  dif.  eq.  (Boole),  93 

Identical  tr.,  4 

Independent  linear  partial  dif.  eqs.,  104 

Infinitesimal  contact  tr.,  185 

characteristic  function  of,  186 

symbol  of,  186 
Infinitesimal  tr.,  6,  29,  197,  215,  218 

distinct,  7 

gr.  generated  by,  TO,  12,  14,  30,  220 

linearly  independent,  143,  217 

r-parameter  gr.  of,  1^6 

symbol  of,  8,  42,  84,  85,  218      • 
Integrating  factor,  37,  47,  69,  76 

common  to  two  dif.  eqs.,  72 

two,  for  the  same  dif.  eq.,  48 
Int>  rmediary  integral,  191 
Invariant,  16,  31 

curve,  17,  18,  31,  32 


247 


248 


INDEX 


Invariant 

differential,  51,  88,  194 

dif.  eq.,  see  Dif.  eq. 

equation,  18,  32 

family  of  curves,  20,  22 

linear  partial  dif.  eq.,  115,  118,  119,  122, 
124 

point,  17,  19,  31 

surface,  31,  32 
Inverse  tr.,  3,  29,  211 
Involute,  70 

of  a  circle,  70 

Involution,  functions  in,  179 
Isothermal  curves,  72,  79,  203 

Jacobian  complete  system,  107,  no 
Jacobi's  identity,  121 

Lie  gr.,  3 

Lie's  principal  theorem,  225 

Lineal  element,  175 

Linear  ordinary  dif.  eq.  of  i.  ord.,  56,  57 

of  2.  ord.,  92,  94,  139,  140,  173,  174 

of  n.  ord.,  102 

Linear  partial  dif.  eq.  i.  u.  a  gr.,  115,  118, 
119 

i.  u.  two  grs.,  122,  124 
Linearly  independent  infl.  trs.,  143,  217 

number  of,  leaving  dif.  eq.  of  ord.  u  ~/  2 

unaltered  limited,  143,  146 
Lines  of  curvature,  81 

Method  of  solution  of 
complete  system,  in,  113 
dif.  eq.  of  i.  ord.,  38,  49,  63,  66,  193 
of  2.  ord.,  88,  134,  137,  165,  169,  193 
of  n.  ord.,  101 

linear  partial  dif.  eq.,  119,  124 
Minimal  lines,  78 

tt-times-extended  gr.,  84  ;  tr.,  83 
Once-extended  gr.,  42 ;  tr.,  41 

Parallel  curves,  70  » 

Path-curve,  4,  ic,  n,  17,  18,  19,  31,  67 
Perspective  tr.,  3 
Point  tr.,  40 

extended,  41,  83,  180 
Poissonian  symbol,  179 
Product  of  trs.,  2 
Projective  tr.,  general,  213 


Reciprocal  polars,  tr.  by,  180,  184 
Riccati  equation,  52,  59,  201 
Rotations,  2 

r-parameter  gr.  of  infl.  trs.,  146 
of  trs.,  211,  214 

Second  differential  invariant,  88 

Separation  of  variables,  63 

Similitudinous  tr.,  3 

Singular  solution,  66 

Subgroup,  149 

Symbol  of  extended  infl.  tr.,  42,  84,  85 

of  infl.  contact  tr.,  186 

of  infl.  tr.,  8,  218 
System,  complete,  see  Complete  system 

Transform  of  a  tr.,  24 
Transformation 

affine,  3,  54 

by  reciprocal  polars,  180,  184 

contact,  178,  181 

extended,  41,  83 

general  projective,  213 

gr.  of,  i,  28,  2ii 

identical,  4 

infl.,  6,  29,  185,  197,  215,  218 

inverse,  3,  29,  211 

perspective,  3 

point,  40 

product  of,  2 

similitudinous,  3 
Translation,  2,  53,  212 
Trivial  gr.,  39,  119,  196 
Twice-extended  gr.,  84;  tr.,  83 
Two-parameter  grs.,  classification  of,  152 
Two-parameter    subgroups    always    exist, 

150 

Types  of  dif.  eqs.  of  i.  ord.  i.  u.  given  grs., 
52,  231 

of  2.  ord.  i.  u.  given  grs.,  90,  236 

of  w.  ord.  i.  u.  given  grs.,  101,  236 

Union  of  elements,  175,  176,  194 

curve  of,  175,  189 
United  elements,  175 

Variables 

canonical.  26,  34,  64,  156 
change  of,  23,  33,  188 
separation  of,  63 


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